
    S_f                       d dl Z d dlmZ d dlmZmZ d dlZd dlZd dl	m
Z
 d dlmZmZmZ d dlmZ d dlmZ d dlmZ d dlmZ d dlmc mZ d d	lmZmZ d
dlmZ d
dlm Z!m"Z# d
dl$m%Z%m&Z&m'Z'm(Z(m)Z)m*Z*m+Z+ d
dl,m-Z-m.Z.m/Z/ d
dl0m1Z1m2Z2m3Z3m4Z4m5Z5m6Z6m7Z7 d
dl8m9Z9 d dl:m;c m<Z< d dl=m>Z> d dl?m@Z@ d dlAm;Z; d ZBd ZCd~dZD G d de'      ZE eEddd      ZF G d de'      ZG eGd ddd      ZH G d  d!e'      ZI eIdd"#      ZJ ej                  d$ej                  z        ZM ej                  eM      ZOd% ZPd& ZQd' ZRd( ZSd) ZTd* ZUd+ ZVd, ZW G d- d.e'      ZX eXd/0      ZY G d1 d2e'      ZZ eZdd3#      Z[ G d4 d5e'      Z\ e\ej                   d6z  ej                  d6z  d7      Z] G d8 d9e'      Z^ e^ddd:      Z_ G d; d<e`      Za G d= d>e@      Zbd? Zcd@ Zd G dA dBe'      Ze eedddC      Zf G dD dEe'      Zg egddF#      Zh G dG dHe'      Zi eidddI      Zj G dJ dKe'      Zk ekddL#      Zl G dM dNe'      Zm emddO#      Zn G dP dQek      Zo eoddR#      Zp G dS dTe'      Zq eqdU0      Zr G dV dWe'      Zs esddX#      Zt G dY dZe'      Zu eudd[#      Zv G d\ d]e'      Zw ewej                   ej                  d^      Zx G d_ d`e'      Zy eyda0      Zz G db dce'      Z{ e{dd0      Z| G de dfe'      Z} e}ddg#      Z~ G dh die'      Z edj0      Zdk Z G dl dme'      Z eddn#      Z G do dpe'      Z eddq#      Z G dr dse'      Z eddt#      Z G du dve'      Z eddw#      Z G dx dye'      Z eddz#      Z G d{ d|e'      Z edd}#      Z G d~ de'      Z edd#      Z G d de'      Z ed0      Z G d de'      Z edd      Z G d de'      Z ed0      Z G d de'      Z edd#      Z G d de'      Z edd#      Z G d de'      Z ed0      Zd Z G d de'      Z edd#      Z G d de      Z edd#      Z G d de'      Z edd#      Z G d de'      Z edd#      Z G d de'      Z ed0      Z G d de'      Z edd#      Zd Z G d de'      Z ed0      Z G d de'      Z ed0      Z G d de'      Z edd#      Z G d de'      Z edd#      Z G d de'      Z edd#      Z G d de'      Z ed0      Z G d de'      Z eddd      Z G d de'      Z edd#      Z G d de'      Z edd¬#      Z G dÄ de'      Z eddŬ#      Z G dƄ de'      Z edȬ0      Z G dɄ de'      Z ed dˬ#      Z G d̄ de'      Z edά0      Z G dτ de'      Z edddѬ      Z G d҄ de'      Z edԬ0      Z G dՄ de'      Z ed׬0      Z G d؄ de'      Z edڬ0      Zdۄ Z G d܄ de'      Z eddެ#      Z G d߄ de'      Z eddᬇ      Z G d de'      Z ed0      Z G d de'      Z ed0      Z G d de'      Z edd#      Zd Z G d de'      Z edd#      Z G d de'      Z edd#      Z G d de'      Z edd#      Z G d de'      Z edd#      Z G d de'      Z ed0      Z G d de'      Z edd#      Z G d de'      Z ed 0      Z G d de'      Z edd#      Zd Z G d de'      Z edd#      Z G d d	e'      Z edd
#      Z G d de'      Z ed0      Z G d de'      Z ed0      Z G d de'      Z edd#      Z G d de'      Z edd#      Z G d de'      Z ed0      Z G d de'      Z eddd      Z G d de'      Z edd#      Z G d  d!e'      Z ed"0      Z G d# d$e'      Z ed%dd&      Z G d' d(e'      Z edd)#      Z  G d* d+e'      Z ed,0      Z ed-0      Z G d. d/e'      Z edd0#      Z G d1 d2e'      Z edd3#      Z G d4 d5e'      Z ed%dd6      Z	 G d7 d8e'      Z
 e
d90      Z G d: d;e'      Z ed<0      Z G d= d>e'      Z eddd?      Z eddd@      Zej"                  r
dAe_         G dB dCe'      Z edddD      Z G dE dFe'      Z eddG#      ZdH ZdI ZdJ Z G dK dLe'      Z edMd
N      Z G dO dPe'      Z eddQ#      Z G dR dSe'      Z edT0      Z G dU dVea      Z G dW dXe'      Z  e dddY      Z! G dZ d[e'      Z" e"d\0      Z# e"ej                   ej                  d]      Z$ G d^ d_e      Z% e%dd`#      Z& G da dbe'      Z' e'dd$ej                  z  dc      Z( G dd dee'      Z) e)df0      Z* G dg dhe'      Z+ e+d di#      Z, G dj dke'      Z- e-dldmn      Z.do Z/ G dp dqe'      Z0 e0drdsddt      Z1 G du dve'      Z2 G dw dxe'      Z3 e3dyd ejh                  z      Z5 G d{ d|e'      Z6 e6dd}#      Z7 e8 e9       ju                         jw                               Z< e%e<e'      \  Z=Z>e=e>z   dvgz   Z?y(      N)Iterable)wrapscached_property
Polynomial)extend_notes_in_docstringreplace_notes_in_docstringinherit_docstring_from)LowLevelCallable)optimize)	integrate)_lazyselect
_lazywhere   )_stats)tukeylambda_variancetukeylambda_kurtosis)get_distribution_names	_kurtosisrv_continuous_skew_get_fixed_fit_value_check_shape
_ShapeInfo)kolmognkolmognpkolmogni)_XMIN_LOGXMIN_EULER_ZETA3_SQRT_PI_SQRT_2_OVER_PI_LOG_SQRT_2_OVER_PI)CensoredData)root_scalar)FitErrorc                     | j                  dd       | j                  dd       | j                  dd       | j                  dd       | rt        d| z        y)a  
    Remove the optimizer-related keyword arguments 'loc', 'scale' and
    'optimizer' from `kwds`.  Then check that `kwds` is empty, and
    raise `TypeError("Unknown arguments: %s." % kwds)` if it is not.

    This function is used in the fit method of distributions that override
    the default method and do not use the default optimization code.

    `kwds` is modified in-place.
    locNscale	optimizermethodzUnknown arguments: %s.)pop	TypeError)kwdss    >lib/python3.12/site-packages/scipy/stats/_continuous_distns.py_remove_optimizer_parametersr1   '   sU     	HHUDHHWdHH[$HHXt04788     c                 .     t                fd       }|S )Nc                    |j                  dd      j                         }t        |t              }|dk(  s|r/|j	                         dkD  rt        t        |       |   |g|i |S |r|j                  } | |g|i |S )Nr,   mlemmr   )	getlower
isinstancer%   num_censoredsupertypefit_uncensored)selfdataargsr/   r,   censoredfuns         r0   wrapperz _call_super_mom.<locals>.wrapper>   s    (E*002dL1T>h4+<+<+>+BdT.tCdCdCC ''tT1D1D11r2   )r   )rC   rD   s   ` r0   _call_super_momrE   :   s"     3Z2 2 Nr2   c                      |xs |dz
  }||z
  } fd} |||      s6|dz  }||z
  }d}t        j                  |      rt        |       |||      s6|S )Nr   c                 r    t        j                   |             t        j                   |            k7  S Nnpsign)lbrackrbrackrC   s     r0   interval_contains_rootz1_get_left_bracket.<locals>.interval_contains_rootV   s(    wws6{#rwws6{';;;r2      zVThe solver could not find a bracket containing a root to an MLE first order condition.)rJ   isinfFitSolverError)rC   rM   rL   diffrN   msgs   `     r0   _get_left_bracketrT   O   sn    !vzFF?D< %VV4	$788F %% %VV4 Mr2   c                   :    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
y	)
	ksone_gena  Kolmogorov-Smirnov one-sided test statistic distribution.

    This is the distribution of the one-sided Kolmogorov-Smirnov (KS)
    statistics :math:`D_n^+` and :math:`D_n^-`
    for a finite sample size ``n >= 1`` (the shape parameter).

    %(before_notes)s

    See Also
    --------
    kstwobign, kstwo, kstest

    Notes
    -----
    :math:`D_n^+` and :math:`D_n^-` are given by

    .. math::

        D_n^+ &= \text{sup}_x (F_n(x) - F(x)),\\
        D_n^- &= \text{sup}_x (F(x) - F_n(x)),\\

    where :math:`F` is a continuous CDF and :math:`F_n` is an empirical CDF.
    `ksone` describes the distribution under the null hypothesis of the KS test
    that the empirical CDF corresponds to :math:`n` i.i.d. random variates
    with CDF :math:`F`.

    %(after_notes)s

    References
    ----------
    .. [1] Birnbaum, Z. W. and Tingey, F.H. "One-sided confidence contours
       for probability distribution functions", The Annals of Mathematical
       Statistics, 22(4), pp 592-596 (1951).

    %(example)s

    c                 >    |dk\  |t        j                  |      k(  z  S Nr   rJ   roundr?   ns     r0   	_argcheckzksone_gen._argcheck       Q1+,,r2   c                 @    t        dddt        j                  fd      gS Nr\   Tr   TFr   rJ   infr?   s    r0   _shape_infozksone_gen._shape_info       3q"&&k=ABBr2   c                 0    t        j                  ||       S rH   )scu	_smirnovpr?   xr\   s      r0   _pdfzksone_gen._pdf   s    a###r2   c                 .    t        j                  ||      S rH   )rh   	_smirnovcrj   s      r0   _cdfzksone_gen._cdf   s    }}Q""r2   c                 .    t        j                  ||      S rH   )scsmirnovrj   s      r0   _sfzksone_gen._sf   s    zz!Qr2   c                 .    t        j                  ||      S rH   )rh   
_smirnovcir?   qr\   s      r0   _ppfzksone_gen._ppf   s    ~~a##r2   c                 .    t        j                  ||      S rH   )rq   smirnovirv   s      r0   _isfzksone_gen._isf       {{1a  r2   N)__name__
__module____qualname____doc__r]   re   rl   ro   rs   rx   r{    r2   r0   rV   rV   f   s,    $J-C$# $!r2   rV                 ?ksone)abnamec                   @    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zy
)	kstwo_gena  Kolmogorov-Smirnov two-sided test statistic distribution.

    This is the distribution of the two-sided Kolmogorov-Smirnov (KS)
    statistic :math:`D_n` for a finite sample size ``n >= 1``
    (the shape parameter).

    %(before_notes)s

    See Also
    --------
    kstwobign, ksone, kstest

    Notes
    -----
    :math:`D_n` is given by

    .. math::

        D_n = \text{sup}_x |F_n(x) - F(x)|

    where :math:`F` is a (continuous) CDF and :math:`F_n` is an empirical CDF.
    `kstwo` describes the distribution under the null hypothesis of the KS test
    that the empirical CDF corresponds to :math:`n` i.i.d. random variates
    with CDF :math:`F`.

    %(after_notes)s

    References
    ----------
    .. [1] Simard, R., L'Ecuyer, P. "Computing the Two-Sided
       Kolmogorov-Smirnov Distribution",  Journal of Statistical Software,
       Vol 39, 11, 1-18 (2011).

    %(example)s

    c                 >    |dk\  |t        j                  |      k(  z  S rX   rY   r[   s     r0   r]   zkstwo_gen._argcheck   r^   r2   c                 @    t        dddt        j                  fd      gS r`   rb   rd   s    r0   re   zkstwo_gen._shape_info   rf   r2   c                 b    dt        |t              s|z  dfS t        j                  |      z  dfS N      ?r   )r9   r   rJ   
asanyarrayr[   s     r0   _get_supportzkstwo_gen._get_support   s;    jH5QL 	2==;KL 	r2   c                     t        ||      S rH   )r   rj   s      r0   rl   zkstwo_gen._pdf   s    1~r2   c                     t        ||      S rH   r   rj   s      r0   ro   zkstwo_gen._cdf   s    q!}r2   c                     t        ||d      S NFcdfr   rj   s      r0   rs   zkstwo_gen._sf   s    q!''r2   c                     t        ||d      S )NTr   r   rv   s      r0   rx   zkstwo_gen._ppf   s    1$''r2   c                     t        ||d      S r   r   rv   s      r0   r{   zkstwo_gen._isf   s    1%((r2   N)r}   r~   r   r   r]   re   r   rl   ro   rs   rx   r{   r   r2   r0   r   r      s1    #H-C(()r2   r   kstwo)momtyper   r   r   c                   4    e Zd ZdZd Zd Zd Zd Zd Zd Z	y)	kstwobign_gena  Limiting distribution of scaled Kolmogorov-Smirnov two-sided test statistic.

    This is the asymptotic distribution of the two-sided Kolmogorov-Smirnov
    statistic :math:`\sqrt{n} D_n` that measures the maximum absolute
    distance of the theoretical (continuous) CDF from the empirical CDF.
    (see `kstest`).

    %(before_notes)s

    See Also
    --------
    ksone, kstwo, kstest

    Notes
    -----
    :math:`\sqrt{n} D_n` is given by

    .. math::

        D_n = \text{sup}_x |F_n(x) - F(x)|

    where :math:`F` is a continuous CDF and :math:`F_n` is an empirical CDF.
    `kstwobign`  describes the asymptotic distribution (i.e. the limit of
    :math:`\sqrt{n} D_n`) under the null hypothesis of the KS test that the
    empirical CDF corresponds to i.i.d. random variates with CDF :math:`F`.

    %(after_notes)s

    References
    ----------
    .. [1] Feller, W. "On the Kolmogorov-Smirnov Limit Theorems for Empirical
       Distributions",  Ann. Math. Statist. Vol 19, 177-189 (1948).

    %(example)s

    c                     g S rH   r   rd   s    r0   re   zkstwobign_gen._shape_info      	r2   c                 .    t        j                  |       S rH   )rh   _kolmogpr?   rk   s     r0   rl   zkstwobign_gen._pdf  s    Qr2   c                 ,    t        j                  |      S rH   )rh   _kolmogcr   s     r0   ro   zkstwobign_gen._cdf  s    ||Ar2   c                 ,    t        j                  |      S rH   )rq   
kolmogorovr   s     r0   rs   zkstwobign_gen._sf  s    }}Qr2   c                 ,    t        j                  |      S rH   )rh   	_kolmogcir?   rw   s     r0   rx   zkstwobign_gen._ppf  s    }}Qr2   c                 ,    t        j                  |      S rH   )rq   kolmogir   s     r0   r{   zkstwobign_gen._isf  s    zz!}r2   N)
r}   r~   r   r   re   rl   ro   rs   rx   r{   r   r2   r0   r   r      s&    #H   r2   r   	kstwobign)r   r   rO   c                 H    t        j                  | dz   dz        t        z  S NrO          @)rJ   exp_norm_pdf_Crk   s    r0   	_norm_pdfr   ,  s     661a4%){**r2   c                 "    | dz   dz  t         z
  S r   )_norm_pdf_logCr   s    r0   _norm_logpdfr   0  s    qD53;''r2   c                 ,    t        j                  |       S rH   )rq   ndtrr   s    r0   	_norm_cdfr   4  s    771:r2   c                 ,    t        j                  |       S rH   )rq   log_ndtrr   s    r0   _norm_logcdfr   8  s    ;;q>r2   c                 ,    t        j                  |       S rH   )rq   ndtrirw   s    r0   	_norm_ppfr   <  s    88A;r2   c                     t        |        S rH   r   r   s    r0   _norm_sfr   @  s    aR=r2   c                     t        |        S rH   r   r   s    r0   _norm_logsfr   D  s    r2   c                     t        |        S rH   r   r   s    r0   	_norm_isfr   H  s    aL=r2   c                       e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 Zd Zd Zd Zd Ze eed      d               Zd Zy)norm_gena  A normal continuous random variable.

    The location (``loc``) keyword specifies the mean.
    The scale (``scale``) keyword specifies the standard deviation.

    %(before_notes)s

    Notes
    -----
    The probability density function for `norm` is:

    .. math::

        f(x) = \frac{\exp(-x^2/2)}{\sqrt{2\pi}}

    for a real number :math:`x`.

    %(after_notes)s

    %(example)s

    c                     g S rH   r   rd   s    r0   re   znorm_gen._shape_infoc  r   r2   Nc                 $    |j                  |      S rH   )standard_normalr?   sizerandom_states      r0   _rvsznorm_gen._rvsf  s    ++D11r2   c                     t        |      S rH   r   r   s     r0   rl   znorm_gen._pdfi  s    |r2   c                     t        |      S rH   r   r   s     r0   _logpdfznorm_gen._logpdfm      Ar2   c                     t        |      S rH   r   r   s     r0   ro   znorm_gen._cdfp      |r2   c                     t        |      S rH   r   r   s     r0   _logcdfznorm_gen._logcdfs  r   r2   c                     t        |      S rH   r   r   s     r0   rs   znorm_gen._sfv  s    {r2   c                     t        |      S rH   )r   r   s     r0   _logsfznorm_gen._logsfy  s    1~r2   c                     t        |      S rH   r   r   s     r0   rx   znorm_gen._ppf|  r   r2   c                     t        |      S rH   r   r   s     r0   r{   znorm_gen._isf  r   r2   c                      y)N)r   r   r   r   r   rd   s    r0   r   znorm_gen._stats      !r2   c                 Z    dt        j                  dt         j                  z        dz   z  S Nr   rO   r   rJ   logpird   s    r0   _entropyznorm_gen._entropy  s"    BFF1RUU7OA%&&r2   a}          For the normal distribution, method of moments and maximum likelihood
        estimation give identical fits, and explicit formulas for the estimates
        are available.
        This function uses these explicit formulas for the maximum likelihood
        estimation of the normal distribution parameters, so the
        `optimizer` and `method` arguments are ignored.

notesc                    |j                  dd       }|j                  dd       }t        |       ||t        d      t        j                  |      }t        j
                  |      j                         st        d      ||j                         }n|}|-t        j                  ||z
  dz  j                               }||fS |}||fS )Nflocfscale3All parameters fixed. There is nothing to optimize.$The data contains non-finite values.rO   )	r-   r1   
ValueErrorrJ   asarrayisfiniteallmeansqrt)r?   r@   r/   r   r   r)   r*   s          r0   r=   znorm_gen.fit  s     xx%(D)$T* 2  ) * * zz${{4 $$&CDD<))+CC>GGdSj1_2245E Ez EEzr2   c                 D    |dz  dk(  rt        j                  |dz
        S y)z
        @returns Moments of standard normal distribution for integer n >= 0

        See eq. 16 of https://arxiv.org/abs/1209.4340v2
        rO   r   r   r   )rq   
factorial2r[   s     r0   _munpznorm_gen._munp  s%     q5A:==Q''r2   NN)r}   r~   r   r   re   r   rl   r   ro   r   rs   r   rx   r{   r   r   rE   r	   r   r=   r   r   r2   r0   r   r   L  sr    ,2"'  6? @@ <	r2   r   norm)r   c                   L    e Zd ZdZej
                  Zd Zd Zd Z	d Z
d Zd Zy)		alpha_gena&  An alpha continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `alpha` ([1]_, [2]_) is:

    .. math::

        f(x, a) = \frac{1}{x^2 \Phi(a) \sqrt{2\pi}} *
                  \exp(-\frac{1}{2} (a-1/x)^2)

    where :math:`\Phi` is the normal CDF, :math:`x > 0`, and :math:`a > 0`.

    `alpha` takes ``a`` as a shape parameter.

    %(after_notes)s

    References
    ----------
    .. [1] Johnson, Kotz, and Balakrishnan, "Continuous Univariate
           Distributions, Volume 1", Second Edition, John Wiley and Sons,
           p. 173 (1994).
    .. [2] Anthony A. Salvia, "Reliability applications of the Alpha
           Distribution", IEEE Transactions on Reliability, Vol. R-34,
           No. 3, pp. 251-252 (1985).

    %(example)s

    c                 @    t        dddt        j                  fd      gS Nr   Fr   FFrb   rd   s    r0   re   zalpha_gen._shape_info      3266{NCDDr2   c                 N    d|dz  z  t        |      z  t        |d|z  z
        z  S Nr   rO   )r   r   r?   rk   r   s      r0   rl   zalpha_gen._pdf  s+    AqDz)A,&y3q5'999r2   c                     dt        j                  |      z  t        |d|z  z
        z   t        j                  t        |            z
  S )Nr   )rJ   r   r   r   r  s      r0   r   zalpha_gen._logpdf  s8    "&&)|l1SU733bffYq\6JJJr2   c                 <    t        |d|z  z
        t        |      z  S Nr   r   r  s      r0   ro   zalpha_gen._cdf  s    3q5!IaL00r2   c           
      b    dt        j                  |t        |t        |      z        z
        z  S r  )rJ   r   r   r   r?   rw   r   s      r0   rx   zalpha_gen._ppf  s(    2::a)AilN";;<<<r2   c                 T    t         j                  gdz  t         j                  gdz  z   S NrO   rJ   rc   nanr?   r   s     r0   r   zalpha_gen._stats  s!    xzRVVHQJ&&r2   N)r}   r~   r   r   r   _open_support_mask_support_maskre   rl   r   ro   rx   r   r   r2   r0   r   r     s4    > "44ME:K1='r2   r   alphac                   :    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
y	)

anglit_gena  An anglit continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `anglit` is:

    .. math::

        f(x) = \sin(2x + \pi/2) = \cos(2x)

    for :math:`-\pi/4 \le x \le \pi/4`.

    %(after_notes)s

    %(example)s

    c                     g S rH   r   rd   s    r0   re   zanglit_gen._shape_info
  r   r2   c                 2    t        j                  d|z        S r  )rJ   cosr   s     r0   rl   zanglit_gen._pdf  s    vvac{r2   c                 Z    t        j                  |t         j                  dz  z         dz  S N   r   rJ   sinr   r   s     r0   ro   zanglit_gen._cdf  s"    vvaai #%%r2   c                 Z    t        j                  |t         j                  dz  z         dz  S r  )rJ   r  r   r   s     r0   rs   zanglit_gen._sf  s"    vva"%%!)m$++r2   c                 z    t        j                  t        j                  |            t         j                  dz  z
  S Nr  )rJ   arcsinr   r   r   s     r0   rx   zanglit_gen._ppf  s&    yy$RUU1W,,r2   c                     dt         j                  t         j                  z  dz  dz
  ddt         j                  dz  dz
  z  t         j                  t         j                  z  dz
  dz  z  fS )	Nr      r   r	  r  `      rO   rJ   r   rd   s    r0   r   zanglit_gen._stats  sR    BEE"%%KN3&RB-?ruuQQR@R-RRRr2   c                 2    dt        j                  d      z
  S Nr   rO   rJ   r   rd   s    r0   r   zanglit_gen._entropy      {r2   N)r}   r~   r   r   re   rl   ro   rs   rx   r   r   r   r2   r0   r  r    s+    &&,-Sr2   r  r  anglitc                   4    e Zd ZdZd Zd Zd Zd Zd Zd Z	y)	arcsine_gena  An arcsine continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `arcsine` is:

    .. math::

        f(x) = \frac{1}{\pi \sqrt{x (1-x)}}

    for :math:`0 < x < 1`.

    %(after_notes)s

    %(example)s

    c                     g S rH   r   rd   s    r0   re   zarcsine_gen._shape_info8  r   r2   c                     t        j                  d      5  dt         j                  z  t        j                  |d|z
  z        z  cd d d        S # 1 sw Y   y xY w)Nignoredivider   r   )rJ   errstater   r   r   s     r0   rl   zarcsine_gen._pdf;  sD    [[) 	.ruu9RWWQ!W--	. 	. 	.s   /AAc                 z    dt         j                  z  t        j                  t        j                  |            z  S Nr   )rJ   r   r#  r   r   s     r0   ro   zarcsine_gen._cdf@  s&    255y2771:...r2   c                 Z    t        j                  t         j                  dz  |z        dz  S r7  r  r   s     r0   rx   zarcsine_gen._ppfC  s"    vvbeeCik"C''r2   c                     d}d}d}d}||||fS )Nr   g      ?r         r   r?   mumu2g1g2s        r0   r   zarcsine_gen._statsF  s$    3Br2   c                      y)Ngοr   rd   s    r0   r   zarcsine_gen._entropyM  s    &r2   N
r}   r~   r   r   re   rl   ro   rx   r   r   r   r2   r0   r/  r/  $  s%    &.
/('r2   r/  arcsinec                       e Zd ZdZd Zy)FitDataErrorz=Raised when input data is inconsistent with fixed parameters.c                 (    d|d|d|df| _         y )Nz>Invalid values in `data`.  Maximum likelihood estimation with z requires that z < (x - loc)/scale  < z for each x in `data`.rA   )r?   distrr8   uppers       r0   __init__zFitDataError.__init__Y  s/    $iui @""'*@B
	r2   Nr}   r~   r   r   rI  r   r2   r0   rD  rD  T  s
    G
r2   rD  c                       e Zd ZdZd Zy)rQ   zN
    Raised when a solver fails to converge while fitting a distribution.
    c                 B    d}||j                  dd      z  }|f| _        y )Nz1Solver for the MLE equations failed to converge: 
 )replacerA   )r?   mesgemsgs      r0   rI  zFitSolverError.__init__g  s%    BT2&&G	r2   NrJ  r   r2   r0   rQ   rQ   a  s    
r2   rQ   c                 t    t        j                  | |z         }||| t        j                  |       z   z  z
  }|S rH   rq   psi)r   r   r\   s1psiabfuncs         r0   _beta_mle_arX  m  s8     FF1q5MEeVbffQi'((DKr2   c                     | \  }}t        j                  ||z         }||| t        j                  |      z   z  z
  ||| t        j                  |      z   z  z
  g}|S rH   rS  )thetar\   rU  s2r   r   rV  rW  s           r0   _beta_mle_abr\  v  sb     DAqFF1q5MEufrvvay())ufrvvay())+DKr2   c                        e Zd ZdZd ZddZd Zd Zd Zd Z	d Z
d	 Zd
 Z fdZe eed       fd              Zd Z xZS )beta_gena  A beta continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `beta` is:

    .. math::

        f(x, a, b) = \frac{\Gamma(a+b) x^{a-1} (1-x)^{b-1}}
                          {\Gamma(a) \Gamma(b)}

    for :math:`0 <= x <= 1`, :math:`a > 0`, :math:`b > 0`, where
    :math:`\Gamma` is the gamma function (`scipy.special.gamma`).

    `beta` takes :math:`a` and :math:`b` as shape parameters.

    %(after_notes)s

    %(example)s

    c                     t        dddt        j                  fd      }t        dddt        j                  fd      }||gS Nr   Fr   r  r   rb   r?   iaibs      r0   re   zbeta_gen._shape_info  ;    UQK@UQK@Bxr2   c                 (    |j                  |||      S rH   beta)r?   r   r   r   r   s        r0   r   zbeta_gen._rvs  s      At,,r2   c                     t        j                  d      5  t        j                  |||      cd d d        S # 1 sw Y   y xY wNr2  over)rJ   r5  _boost	_beta_pdfr?   rk   r   r   s       r0   rl   zbeta_gen._pdf  s7     [[h' 	-##Aq!,	- 	- 	-	   8Ac                     t        j                  |dz
  |       t        j                  |dz
  |      z   }|t        j                  ||      z  }|S r  )rq   xlog1pyxlogybetaln)r?   rk   r   r   lPxs        r0   r   zbeta_gen._logpdf  sE    jjS1"%S!(<<ryyA
r2   c                 0    t        j                  |||      S rH   )rl  	_beta_cdfrn  s       r0   ro   zbeta_gen._cdf  s    1a((r2   c                 0    t        j                  |||      S rH   )rl  _beta_sfrn  s       r0   rs   zbeta_gen._sf  s    q!Q''r2   c                 0    t        j                  |||      S rH   )rq   betainccinvrn  s       r0   r{   zbeta_gen._isf  s    ~~aA&&r2   c                 0    t        j                  |||      S rH   )rq   
betaincinvr?   rw   r   r   s       r0   rx   zbeta_gen._ppf  s    }}Q1%%r2   c                     t        j                  ||      t        j                  ||      t        j                  ||      t        j                  ||      fS rH   )rl  
_beta_mean_beta_variance_beta_skewness_beta_kurtosis_excessr?   r   r   s      r0   r   zbeta_gen._stats  sL    a#!!!Q'!!!Q'((A.	0 	0r2   c                     t        |t              r|j                         }t        |      t	        |      fd}t        j                  |d      \  }}t        | !  |||f      S )Nc                 J   | \  }}d||z
  z  t        j                  ||z   dz         z  ||z   dz   z  t        j                  ||z        z  }|dz  |dz  d|z  dz
  z  z
  |dz  |dz   z  z   d|z  |z  |dz   z  z
  }|||z  ||z   dz   z  ||z   dz   z  z  }|dz  }|z
  |z
  gS )NrO   r         rJ   r   )rk   r   r   skkur>  r?  s        r0   rW  z beta_gen._fitstart.<locals>.func  s    DAqAaCQ++q1uqy9BGGAaCLHBA1ac!e$q!tQqSz1AaCE1Q3K?B!A#qs1u+qs1u%%B!GBrE2b5>!r2   )r   r   rF  )	r9   r%   	_uncensorr   r   r   fsolver;   	_fitstart)r?   r@   rW  r   r   r>  r?  	__class__s        @@r0   r  zbeta_gen._fitstart  s_    dL)>>#D4[t_	" tZ01w QF 33r2   z        In the special case where `method="MLE"` and
        both `floc` and `fscale` are given, a
        `ValueError` is raised if any value `x` in `data` does not satisfy
        `floc < x < floc + fscale`.

r   c           	         |j                  dd       }|j                  dd       }||t        |   |g|i |S |j                  dd        |j                  dd        t	        |g d      }t	        |g d      }t        |       ||t        d      t        j                  |      j                         st        d      t        j                  |      |z
  |z  }t        j                  |dk        st        j                  |dk\        rt        d	|||z   
      |j                         }||||}	d|z
  }d|z
  }n|}	|	|z  d|z
  z  }
t        j                  t         |
|	t#        |      t        j$                  |      j'                         fd      \  }}}}|dk7  rt)        |      |d   }
||	|
}	}
nt        j$                  |      j'                         }t+        j,                  |       j'                         }|d|z
  z  |j/                  d      z  dz
  }||z  }
d|z
  |z  }	t        j                  t0        |
|	gt#        |      ||fd      \  }}}}|dk7  rt)        |      |\  }
}	|
|	||fS )Nr   r   f0fafix_a)f1fbfix_br   r   r   r   rg  r8   rH  T)rA   full_output)rP  )ddof)r7   r;   r=   r-   r   r1   r   rJ   r   r   ravelanyrD  r   r   r  rX  lenr   sumrQ   rq   log1pvarr\  )r?   r@   rA   r/   r   r   r  r  xbarr   r   rZ  infoierrP  rU  r[  facr  s                     r0   r=   zbeta_gen.fit  s    xx%(D)<6>7;t3d3d33 	4 !$(=>!$(=>$T*>bn ) * * {{4 $$&CDD %/66$!)tqy 1vTGGyy{>R^ ~ 4x4x DAH%A &.__QTBFF4L$4$4$67 &"E4d
 ax$$//aA~ !1 !!#B4%$$&B !d(#dhhAh&66:Cs
ATS A &.__q!f$iR( &"E4d
 ax$$//DAq!T6!!r2   c                     d }d }d }d }|dk\  r|dk\  r	 |||      S |dk  r||z
  dk\  r| ||      k\  r	 |||      S |dk  r||z
  dk\  r| ||      k\  r	 |||      S  |||      S )Nc                     t        j                  | |      | dz
  t        j                  |       z  z
  |dz
  t        j                  |      z  z
  | |z   dz
  t        j                  | |z         z  z   S r*  )rq   rs  rT  r   r   s     r0   regularz"beta_gen._entropy.<locals>.regular@  sf    IIaOq1uq	&99UbffQi'(+,q519q1u*EF Gr2   c                    | |z   }dt        j                  dt         j                  z        t        j                  |       z   t        j                  |      z   dt        j                  |      z  z
  dz   z  }d|z  d|dz  z  z   |dz  z   d|d	z  z  z
  }d
| z  d| dz  z  z
  | dz  z
  | d	z  z   }d
|z  d|dz  z  z
  |dz  z
  |d	z  z   }|||z   |z   dz  z   S )Nr   rO   r  r   n                         i
   x   r   )r   r   sum_ablog_termt1t2t3s          r0   asymptotic_ab_largez.beta_gen._entropy.<locals>.asymptotic_ab_largeD  s    UFqw"&&)+bffQi7!BFF6N:JJQNH Vbo-<q~MBQAtG#ag-47BQAtG#ag-47BrBw|s222r2   c                    | |z   }t        j                  |       | dz
  t        j                  |       z  z
  }dd|z  z  dd|z  z  z   |dz  dz  z
  |dz  dz  z
  |dz  dz  z   |d	z  d
z  z   |dz  d
z  z
  d|z  z   dd|z  z  z
  |dz  dz  z   |dz  dz  z   |dz  dz  z
  |d	z  d
z  z
  |dz  dz  z   }|t        j                  | |z        z  t        j
                  |      z   dt        j
                  |      z  z
  }||z   |z   S )Nr   rO      r  r  r  r                 r  <   ~   )rq   gammalnrT  rJ   r  r   )r   r   r  r  r  r  s         r0   asymptotic_b_largez-beta_gen._entropy.<locals>.asymptotic_b_largeN  sM   UFA!a%266!9!44BQqS	Ar!tH$q$wrz1AtGCK?!T'#+MT'#+ !4,./h79:BvIG$,q.!#)4<#346<dl2oF $,s"# &,T\#%56  bhhqsm+bffQi7!BFF6N:JJH7X%%r2   c                     | dk(  ryt        j                  |       }t        |      }t        | d|z  z        dz   }|dd|z   z  z  S )Nr   i  r  rO      )rJ   log10int)vjdigitsds       r0   threshold_largez*beta_gen._entropy.<locals>.threshold_largeZ  sL    CxAVFAf$%)AR!a%[= r2   g    RAg    (RAg    .Ar   )r?   r   r   r  r  r  r  s          r0   r   zbeta_gen._entropy?  s    	G	3
	&	! ;1;&q!,,%ZAESLQ/!2D-D%a++%ZAESLQ/!2D-D%a++1a= r2   r   )r}   r~   r   r   re   r   rl   r   ro   rs   r{   rx   r   r  rE   r   r   r=   r   __classcell__r  s   @r0   r^  r^    sj    .
--
)('&04" } 5+ ,
c", c"J+!r2   r^  rg  c                   Z    e Zd ZdZej
                  Zd ZddZd Z	d Z
d Zd Zd	 Zd
 Zy)betaprime_gena  A beta prime continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `betaprime` is:

    .. math::

        f(x, a, b) = \frac{x^{a-1} (1+x)^{-a-b}}{\beta(a, b)}

    for :math:`x >= 0`, :math:`a > 0`, :math:`b > 0`, where
    :math:`\beta(a, b)` is the beta function (see `scipy.special.beta`).

    `betaprime` takes ``a`` and ``b`` as shape parameters.

    The distribution is related to the `beta` distribution as follows:
    If :math:`X` follows a beta distribution with parameters :math:`a, b`,
    then :math:`Y = X/(1-X)` has a beta prime distribution with
    parameters :math:`a, b` ([1]_).

    The beta prime distribution is a reparametrized version of the
    F distribution.  The beta prime distribution with shape parameters
    ``a`` and ``b`` and ``scale = s`` is equivalent to the F distribution
    with parameters ``d1 = 2*a``, ``d2 = 2*b`` and ``scale = (a/b)*s``.
    For example,

    >>> from scipy.stats import betaprime, f
    >>> x = [1, 2, 5, 10]
    >>> a = 12
    >>> b = 5
    >>> betaprime.pdf(x, a, b, scale=2)
    array([0.00541179, 0.08331299, 0.14669185, 0.03150079])
    >>> f.pdf(x, 2*a, 2*b, scale=(a/b)*2)
    array([0.00541179, 0.08331299, 0.14669185, 0.03150079])

    %(after_notes)s

    References
    ----------
    .. [1] Beta prime distribution, Wikipedia,
           https://en.wikipedia.org/wiki/Beta_prime_distribution

    %(example)s

    c                     t        dddt        j                  fd      }t        dddt        j                  fd      }||gS r`  rb   ra  s      r0   re   zbetaprime_gen._shape_info  rd  r2   Nc                 l    t         j                  |||      }t         j                  |||      }||z  S Nr   r   )gammarvs)r?   r   r   r   r   u1u2s          r0   r   zbetaprime_gen._rvs  s3    YYqt,Y?YYqt,Y?Bwr2   c                 N    t        j                  | j                  |||            S rH   rJ   r   r   rn  s       r0   rl   zbetaprime_gen._pdf      vvdll1a+,,r2   c                     t        j                  |dz
  |      t        j                  ||z   |      z
  t        j                  ||      z
  S r  )rq   rr  rq  rs  rn  s       r0   r   zbetaprime_gen._logpdf  s:    xxC#bjjQ&::RYYq!_LLr2   c                 0    t        |dkD  |||gd d       S )Nr   c                 <    t         j                  dd| z   z  ||      S rX   rg  rs   x_a_b_s      r0   <lambda>z$betaprime_gen._cdf.<locals>.<lambda>  s    txx1R4"b9 r2   c                 <    t         j                  | d| z   z  ||      S rX   rg  ro   r  s      r0   r  z$betaprime_gen._cdf.<locals>.<lambda>  s    $))B"Ir2"> r2   f2r   rn  s       r0   ro   zbetaprime_gen._cdf  s(     EAq!99>@ 	@r2   c                 0    t        |dkD  |||gd d       S )Nr   c                 <    t         j                  dd| z   z  ||      S rX   r  r  s      r0   r  z#betaprime_gen._sf.<locals>.<lambda>  s    tyyAbD2r: r2   c                 <    t         j                  | d| z   z  ||      S rX   r  r  s      r0   r  z#betaprime_gen._sf.<locals>.<lambda>  s    $((2qt9b""= r2   r  r  rn  s       r0   rs   zbetaprime_gen._sf  s$    EAq!9:=
 	
r2   c                 T   t        j                  |||      \  }}}t        j                  j	                  |||      }t        j
                  d      5  |d|z
  z  }d d d        |dkD  }dt        j                  j                  ||   ||   ||         z  dz
  |<   |S # 1 sw Y   CxY w)Nr2  r3  r   gH.?)rJ   broadcast_arraysstatsrg  rx   r5  r{   )r?   pr   r   routis          r0   rx   zbetaprime_gen._ppf  s    %%aA.1a JJOOAq!$[[) 	q1u+C	Z5::??1Q41qt44q8A
		 	s   	BB'c                 N    t        |kD  ||ffdt        j                        S )Nc                     t        j                  t        ddz         D cg c]  }| |z   dz
  ||z
  z   c}d      S c c}w )Nr   r   axis)rJ   prodrange)r   r   r  r\   s      r0   r  z%betaprime_gen._munp.<locals>.<lambda>  s:    q!A#!GA1Q3q51Q3-!GaP !Gs   A 	fillvaluer   rJ   rc   )r?   r\   r   r   s    `  r0   r   zbetaprime_gen._munp  s'    EAq6Pff 	r2   r   )r}   r~   r   r   r   r  r  re   r   rl   r   ro   rs   rx   r   r   r2   r0   r  r  p  s?    .^ "44M

-M@
r2   r  	betaprimec                   6    e Zd ZdZd Zd Zd Zd Zd	dZd Z	y)
bradford_genab  A Bradford continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `bradford` is:

    .. math::

        f(x, c) = \frac{c}{\log(1+c) (1+cx)}

    for :math:`0 <= x <= 1` and :math:`c > 0`.

    `bradford` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                 @    t        dddt        j                  fd      gS NcFr   r  rb   rd   s    r0   re   zbradford_gen._shape_info  r  r2   c                 D    |||z  dz   z  t        j                  |      z  S r  rq   r  r?   rk   r  s      r0   rl   zbradford_gen._pdf  s!    AaC#I!,,r2   c                 ^    t        j                  ||z        t        j                  |      z  S rH   r   r  s      r0   ro   zbradford_gen._cdf  s!    xx!}rxx{**r2   c                 ^    t        j                  |t        j                  |      z        |z  S rH   rq   expm1r  r?   rw   r  s      r0   rx   zbradford_gen._ppf  s"    xxBHHQK(1,,r2   c                 j   t        j                  d|z         }||z
  ||z  z  }|dz   |z  d|z  z
  d|z  |z  |z  z  }d }d }d|v r{t        j                  d      d|z  |z  d|z  |z  |dz   z  z
  d|z  |z  ||dz   z  dz   z  z   z  }|t        j                  |||dz
  z  d|z  z   z        d|z  |dz
  z  d|z  z   z  z  }d	|v rj|dz  |dz
  z  |d|z  d
z
  z  dz   z  d|z  |z  |z  |dz
  z  |dz
  z  z   d|z  |z  |z  d|z  dz
  z  z   d|dz  z  z   }|d|z  ||dz
  z  d|z  z   dz  z  z  }||||fS )Nr   r   rO   sr  	   r  r  kr%     r     )rJ   r   r   )r?   r  momentsr
  r<  r=  r>  r?  s           r0   r   zbradford_gen._stats  s   FF3q5McAaC[#qyQ1Qq)'>RT!VAaCE1Q3K/!Aq!A#wqy0AABB"''!Q!WQqS[/*AaC1IacM::B'>Q$!*a1Rjm,RT!VAXqs^QqS-AAA#a%'1Q3r6"#%'1W-B!A#q!A#wqs{Q&&&B3Br2   c                 n    t        j                  d|z         }|dz  t        j                  ||z        z
  S Nr   r   r+  )r?   r  r
  s      r0   r   zbradford_gen._entropy  s.    FF1Q3Kurvvac{""r2   NmvrA  r   r2   r0   r  r    s&    *E-+-#r2   r  bradfordc                   R    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zd Zd Zy)burr_gena  A Burr (Type III) continuous random variable.

    %(before_notes)s

    See Also
    --------
    fisk : a special case of either `burr` or `burr12` with ``d=1``
    burr12 : Burr Type XII distribution
    mielke : Mielke Beta-Kappa / Dagum distribution

    Notes
    -----
    The probability density function for `burr` is:

    .. math::

        f(x; c, d) = c d \frac{x^{-c - 1}}
                              {{(1 + x^{-c})}^{d + 1}}

    for :math:`x >= 0` and :math:`c, d > 0`.

    `burr` takes ``c`` and ``d`` as shape parameters for :math:`c` and
    :math:`d`.

    This is the PDF corresponding to the third CDF given in Burr's list;
    specifically, it is equation (11) in Burr's paper [1]_. The distribution
    is also commonly referred to as the Dagum distribution [2]_. If the
    parameter :math:`c < 1` then the mean of the distribution does not
    exist and if :math:`c < 2` the variance does not exist [2]_.
    The PDF is finite at the left endpoint :math:`x = 0` if :math:`c * d >= 1`.

    %(after_notes)s

    References
    ----------
    .. [1] Burr, I. W. "Cumulative frequency functions", Annals of
       Mathematical Statistics, 13(2), pp 215-232 (1942).
    .. [2] https://en.wikipedia.org/wiki/Dagum_distribution
    .. [3] Kleiber, Christian. "A guide to the Dagum distributions."
       Modeling Income Distributions and Lorenz Curves  pp 97-117 (2008).

    %(example)s

    c                     t        dddt        j                  fd      }t        dddt        j                  fd      }||gS Nr  Fr   r  r  rb   r?   icids      r0   re   zburr_gen._shape_infoJ  rd  r2   c                 \    t        |dk(  |||gd d       }|j                  dk(  r|d   S |S )Nr   c                 6    ||z  | ||z  dz
  z  z  d| |z  z   z  S rX   r   r  c_d_s      r0   r  zburr_gen._pdf.<locals>.<lambda>S  s(    rBw"r"uQw-8ABJG r2   c                 @    ||z  | | dz
  z  z  d| | z  z   |dz   z  z  S Nr   r   r   r  s      r0   r  zburr_gen._pdf.<locals>.<lambda>T  s6    27bbS3Y.?#@%&_"s($C$E r2   r  r   r   ndimr?   rk   r  r  outputs        r0   rl   zburr_gen._pdfO  sB    FQ1IGFG
 ;;!":r2   c                 \    t        |dk(  |||gd d       }|j                  dk(  r|d   S |S )Nr   c                     t        j                  |      t        j                  |      z   t        j                  ||z  dz
  |       z   |dz   t        j                  | |z        z  z
  S rX   )rJ   r   rq   rr  r  r  s      r0   r  z"burr_gen._logpdf.<locals>.<lambda>]  sQ    r
RVVBZ 7"((2b519b:Q Q#%a4288BH+="=!> r2   c                     t        j                  |      t        j                  |      z   t        j                  | dz
  |       z   t        j                  |dz   | | z        z
  S rX   rJ   r   rq   rr  rq  r  s      r0   r  z"burr_gen._logpdf.<locals>.<lambda>_  sR    266":r
#:%'XXrcAgr%:$;%'ZZ1bB3i%@$A r2   r  r   r!  r#  s        r0   r   zburr_gen._logpdfZ  sD    FQ1I?B	C ;;!":r2   c                     d|| z  z   | z  S rX   r   r?   rk   r  r  s       r0   ro   zburr_gen._cdff  s    AGr""r2   c                 <    t        j                  || z        | z  S rH   r   r*  s       r0   r   zburr_gen._logcdfi  s    xxQB QB''r2   c                 N    t        j                  | j                  |||            S rH   rJ   r   r   r*  s       r0   rs   zburr_gen._sfl      vvdkk!Q*++r2   c                 D    t        j                  d|| z  z   | z         S rX   rJ   r  r*  s       r0   r   zburr_gen._logsfo  s%    xx1qA2w;1"--..r2   c                 $    |d|z  z  dz
  d|z  z  S N      r   r   r?   rw   r  r  s       r0   rx   zburr_gen._ppfr  s    DFa46**r2   c                 l    t        j                  d|z  |       }t        j                  |      d|z  z  S Nr3  rq   rq  r  )r?   rw   r  r  _qs        r0   r{   zburr_gen._isfu  s/    ZZq1"%xx|q))r2   c           	         t        j                  dd      j                  dd      |z  }t        j                  ||z   d|z
        |z  \  }}}}t        j
                  |dkD  |t         j                        }||dz  z
  }	t        j
                  |dkD  |	t         j                        }
t        |dkD  |||||	fd t         j                  	      }t        |d
kD  ||||||	fd t         j                  	      }t        j                  |      dk(  r>|j                         |
j                         |j                         |j                         fS ||
||fS )Nr      r  r   rO   r         @c                 \    |d|z  |z  z
  d|dz  z  z   t        j                  |dz        z  S )Nr  rO   r  )r  e1e2e3mu2_if_cs        r0   r  z!burr_gen._stats.<locals>.<lambda>  s5    b1R47lQr1uW.D/1ww1}/E.F r2   r        @c                 T    |d|z  |z  z
  d|z  |dz  z  z   d|dz  z  z
  |dz  z  dz
  S )Nr  r  rO   r  r   )r  r=  r>  r?  e4r@  s         r0   r  z!burr_gen._stats.<locals>.<lambda>  sB    qtBw,2b!e+aAg51DI r2   r   )
rJ   arangereshaperq   rg  wherer  r   r"  item)r?   r  r  ncr=  r>  r?  rC  r<  r@  r=  r>  r?  s                r0   r   zburr_gen._statsy  s+   YYq!_$$Qq)A-Rb1A5BBXXa#gr266*A:hhq3w"&&1GBH%Gff GBB)Kff 771:?779chhj"'')RWWY>>3Br2   c                     d t        j                  |      t        j                  |      t        j                  |      }}}t        ||kD  ||k(  z  ||k(  z  |||ffdt         j                        S )Nc                 P    d| z  |z  }|t        j                  d|z
  ||z         z  S r  rq   rg  r\   r  r  rH  s       r0   __munpzburr_gen._munp.<locals>.__munp  -    a!BrwwsRxR000r2   c                      || |      S rH   r   )r  r  r\   _burr_gen__munps      r0   r  z burr_gen._munp.<locals>.<lambda>  s    &Aq/ r2   )rJ   r   r   r  )r?   r\   r  r  rP  s       @r0   r   zburr_gen._munp  sf    	1 **Q-A

1a11q5Q!V,Q7!Q9&&" 	"r2   N)r}   r~   r   r   re   rl   r   ro   r   rs   r   rx   r{   r   r   r   r2   r0   r  r    s?    +`
	
#(,/+*."r2   r  burrc                   F    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zy)
burr12_gena}  A Burr (Type XII) continuous random variable.

    %(before_notes)s

    See Also
    --------
    fisk : a special case of either `burr` or `burr12` with ``d=1``
    burr : Burr Type III distribution

    Notes
    -----
    The probability density function for `burr12` is:

    .. math::

        f(x; c, d) = c d \frac{x^{c-1}}
                              {(1 + x^c)^{d + 1}}

    for :math:`x >= 0` and :math:`c, d > 0`.

    `burr12` takes ``c`` and ``d`` as shape parameters for :math:`c`
    and :math:`d`.

    This is the PDF corresponding to the twelfth CDF given in Burr's list;
    specifically, it is equation (20) in Burr's paper [1]_.

    %(after_notes)s

    The Burr type 12 distribution is also sometimes referred to as
    the Singh-Maddala distribution from NIST [2]_.

    References
    ----------
    .. [1] Burr, I. W. "Cumulative frequency functions", Annals of
       Mathematical Statistics, 13(2), pp 215-232 (1942).

    .. [2] https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/b12pdf.htm

    .. [3] "Burr distribution",
       https://en.wikipedia.org/wiki/Burr_distribution

    %(example)s

    c                     t        dddt        j                  fd      }t        dddt        j                  fd      }||gS r  rb   r  s      r0   re   zburr12_gen._shape_info  rd  r2   c                 N    t        j                  | j                  |||            S rH   r  r*  s       r0   rl   zburr12_gen._pdf  r  r2   c                     t        j                  |      t        j                  |      z   t        j                  |dz
  |      z   t        j                  | dz
  ||z        z   S rX   r(  r*  s       r0   r   zburr12_gen._logpdf  sL    vvay266!9$rxxAq'99BJJr!tQPQT<RRRr2   c                 P    t        j                  | j                  |||             S rH   rq   r  r   r*  s       r0   ro   zburr12_gen._cdf  !    Q1-...r2   c                 B    t        j                  d||z  z   | z         S rX   r   r*  s       r0   r   zburr12_gen._logcdf  s#    xx!ad(qb))**r2   c                 N    t        j                  | j                  |||            S rH   r-  r*  s       r0   rs   zburr12_gen._sf  r.  r2   c                 6    t        j                  | ||z        S rH   rq   rq  r*  s       r0   r   zburr12_gen._logsf  s    zz1"ad##r2   c                 l    t        j                  d|z  t        j                  |       z        d|z  z  S )Nr  r   r  r4  s       r0   rx   zburr12_gen._ppf  s/     xx1rxx|+,qs33r2   c                 T    d }t        ||z  |kD  |||f|t        j                        S )Nc                 P    d| z  |z  }|t        j                  d|z   ||z
        z  S r  rK  rL  s       r0   moment_if_existsz*burr12_gen._munp.<locals>.moment_if_exists  rN  r2   r  )r   rJ   r  )r?   r\   r  r  ra  s        r0   r   zburr12_gen._munp  s2    	1 !a%!)aAY0@$&FF, 	,r2   N)r}   r~   r   r   re   rl   r   ro   r   rs   r   rx   r   r   r2   r0   rS  rS    s6    +X
-S/+,$4,r2   rS  burr12c                   X    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zd Zd Zd Zy)fisk_gena  A Fisk continuous random variable.

    The Fisk distribution is also known as the log-logistic distribution.

    %(before_notes)s

    See Also
    --------
    burr

    Notes
    -----
    The probability density function for `fisk` is:

    .. math::

        f(x, c) = \frac{c x^{c-1}}
                       {(1 + x^c)^2}

    for :math:`x >= 0` and :math:`c > 0`.

    Please note that the above expression can be transformed into the following
    one, which is also commonly used:

    .. math::

        f(x, c) = \frac{c x^{-c-1}}
                       {(1 + x^{-c})^2}

    `fisk` takes ``c`` as a shape parameter for :math:`c`.

    `fisk` is a special case of `burr` or `burr12` with ``d=1``.

    Suppose ``X`` is a logistic random variable with location ``l``
    and scale ``s``. Then ``Y = exp(X)`` is a Fisk (log-logistic)
    random variable with ``scale = exp(l)`` and shape ``c = 1/s``.

    %(after_notes)s

    %(example)s

    c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zfisk_gen._shape_info  r  r2   c                 0    t         j                  ||d      S r  )rQ  rl   r  s      r0   rl   zfisk_gen._pdf"  s    yyAs##r2   c                 0    t         j                  ||d      S r  )rQ  ro   r  s      r0   ro   zfisk_gen._cdf&      yyAs##r2   c                 0    t         j                  ||d      S r  )rQ  rs   r  s      r0   rs   zfisk_gen._sf)  s    xx1c""r2   c                 0    t         j                  ||d      S r  )rQ  r   r  s      r0   r   zfisk_gen._logpdf,  s    ||Aq#&&r2   c                 0    t         j                  ||d      S r  )rQ  r   r  s      r0   r   zfisk_gen._logcdf0  s    ||Aq#&&r2   c                 0    t         j                  ||d      S r  )rQ  r   r  s      r0   r   zfisk_gen._logsf3  s    {{1a%%r2   c                 0    t         j                  ||d      S r  )rQ  rx   r  s      r0   rx   zfisk_gen._ppf6  rh  r2   c                 0    t         j                  ||d      S r  )rQ  r{   r  s      r0   r{   zfisk_gen._isf9  rh  r2   c                 0    t         j                  ||d      S r  )rQ  r   r?   r\   r  s      r0   r   zfisk_gen._munp<  s    zz!Q$$r2   c                 .    t         j                  |d      S r  )rQ  r   r?   r  s     r0   r   zfisk_gen._stats?  s    {{1c""r2   c                 2    dt        j                  |      z
  S r  r+  rr  s     r0   r   zfisk_gen._entropyB      266!9}r2   N)r}   r~   r   r   re   rl   ro   rs   r   r   r   rx   r{   r   r   r   r   r2   r0   rd  rd    sE    )TE$$#''&$$%#r2   rd  fiskc                   H    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 ZddZy
)
cauchy_gena	  A Cauchy continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `cauchy` is

    .. math::

        f(x) = \frac{1}{\pi (1 + x^2)}

    for a real number :math:`x`.

    %(after_notes)s

    %(example)s

    c                     g S rH   r   rd   s    r0   re   zcauchy_gen._shape_info]  r   r2   c                 :    dt         j                  z  d||z  z   z  S r  r(  r   s     r0   rl   zcauchy_gen._pdf`      255y#ac'""r2   c                 Z    ddt         j                  z  t        j                  |      z  z   S r   rJ   r   arctanr   s     r0   ro   zcauchy_gen._cdfd  "    SYryy|+++r2   c                 v    t        j                  t         j                  |z  t         j                  dz  z
        S r7  rJ   tanr   r   s     r0   rx   zcauchy_gen._ppfg  s&    vvbeeAgbeeCi'((r2   c                 Z    ddt         j                  z  t        j                  |      z  z
  S r   r|  r   s     r0   rs   zcauchy_gen._sfj  r~  r2   c                 v    t        j                  t         j                  dz  t         j                  |z  z
        S r7  r  r   s     r0   r{   zcauchy_gen._isfm  s&    vvbeeCia'((r2   c                 ~    t         j                  t         j                  t         j                  t         j                  fS rH   rJ   r  rd   s    r0   r   zcauchy_gen._statsp  !    vvrvvrvvrvv--r2   c                 N    t        j                  dt         j                  z        S r"  r   rd   s    r0   r   zcauchy_gen._entropys      vvagr2   Nc                     t        |t              r|j                         }t        j                  |g d      \  }}}|||z
  dz  fS )N   2   K   rO   r9   r%   r  rJ   
percentile)r?   r@   rA   p25p50p75s         r0   r  zcauchy_gen._fitstartv  sA    dL)>>#DdL9S#S3YM!!r2   rH   )r}   r~   r   r   re   rl   ro   rx   rs   r{   r   r   r  r   r2   r0   rw  rw  I  s4    &#,),)."r2   rw  cauchyc                   N    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 Zd Zd Zy)chi_gena  A chi continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `chi` is:

    .. math::

        f(x, k) = \frac{1}{2^{k/2-1} \Gamma \left( k/2 \right)}
                   x^{k-1} \exp \left( -x^2/2 \right)

    for :math:`x >= 0` and :math:`k > 0` (degrees of freedom, denoted ``df``
    in the implementation). :math:`\Gamma` is the gamma function
    (`scipy.special.gamma`).

    Special cases of `chi` are:

        - ``chi(1, loc, scale)`` is equivalent to `halfnorm`
        - ``chi(2, 0, scale)`` is equivalent to `rayleigh`
        - ``chi(3, 0, scale)`` is equivalent to `maxwell`

    `chi` takes ``df`` as a shape parameter.

    %(after_notes)s

    %(example)s

    c                 @    t        dddt        j                  fd      gS NdfFr   r  rb   rd   s    r0   re   zchi_gen._shape_info      4BFF^DEEr2   Nc                 X    t        j                  t        j                  |||            S r  )rJ   r   chi2r  r?   r  r   r   s       r0   r   zchi_gen._rvs  s     wwtxxLxIJJr2   c                 L    t        j                  | j                  ||            S rH   r  r?   rk   r  s      r0   rl   zchi_gen._pdf  s     vvdll1b)**r2   c                     t        j                  d      dt        j                  d      z  |z  z
  t        j                  d|z        z
  }|t        j                  |dz
  |      z   d|dz  z  z
  S )NrO   r   r   )rJ   r   rq   r  rr  )r?   rk   r  ls       r0   r   zchi_gen._logpdf  s]    FF1I266!9R'"**RU*;;288BGQ''"QT'11r2   c                 @    t        j                  d|z  d|dz  z        S Nr   rO   rq   gammaincr  s      r0   ro   zchi_gen._cdf  s    {{2b5"QT'**r2   c                 @    t        j                  d|z  d|dz  z        S r  rq   	gammainccr  s      r0   rs   zchi_gen._sf  s    ||BrE2ad7++r2   c                 `    t        j                  dt        j                  d|z  |      z        S NrO   r   rJ   r   rq   gammaincinvr?   rw   r  s      r0   rx   zchi_gen._ppf  s%    wwq2q1122r2   c                 `    t        j                  dt        j                  d|z  |      z        S r  rJ   r   rq   gammainccinvr  s      r0   r{   zchi_gen._isf  s%    wwqB2233r2   c                 z   t        j                  d      t        j                  d|z  d      z  }|||z  z
  }d|dz  z  |dd|z  z
  z  z   t        j                  t        j
                  |d            z  }d|z  d|z
  z  d|dz  z  z
  d|dz  z  d|z  dz
  z  z   }|t        j                  |d	z        z  }||||fS )
NrO   r   r;  r         ?r   r  r  r   )rJ   r   rq   pochr   powerr?   r  r<  r=  r>  r?  s         r0   r   zchi_gen._stats  s    WWQZ"''#(C002b5jCi"a"f+%rzz"((32D'EErT3r6]1RU7"Qr1uW"Q%77
bjjc""3Br2   c                 4    d }d }t        |dk  |f||      S )Nc                     t        j                  d| z        d| t        j                  d      z
  | dz
  t        j                  d| z        z  z
  z  z   S r   )rq   r  rJ   r   digammar  s    r0   regular_formulaz)chi_gen._entropy.<locals>.regular_formula  sM    JJrBw'R"&&)^rAvC"H9M.MMNO Pr2   c                     dt        j                  t         j                        dz  z   | dz  dz  z
  | dz  dz  z
  d| dz  z  z
  | dz  d	z  z   S )
Nr   rO   r  r  r	  gll?   r   r  s    r0   asymptotic_formulaz,chi_gen._entropy.<locals>.asymptotic_formula  sY    "&&-/)RVQJ6"b&!CBFm$')2vrk2 3r2   g     r@r  r  )r?   r  r  r  s       r0   r   zchi_gen._entropy  s+    	P	3 "s(RFO/1 	1r2   r   r}   r~   r   r   re   r   rl   r   ro   rs   rx   r{   r   r   r   r2   r0   r  r    s;    <FK+2+,341r2   r  chic                   N    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 Zd Zd Zy)chi2_gena  A chi-squared continuous random variable.

    For the noncentral chi-square distribution, see `ncx2`.

    %(before_notes)s

    See Also
    --------
    ncx2

    Notes
    -----
    The probability density function for `chi2` is:

    .. math::

        f(x, k) = \frac{1}{2^{k/2} \Gamma \left( k/2 \right)}
                   x^{k/2-1} \exp \left( -x/2 \right)

    for :math:`x > 0`  and :math:`k > 0` (degrees of freedom, denoted ``df``
    in the implementation).

    `chi2` takes ``df`` as a shape parameter.

    The chi-squared distribution is a special case of the gamma
    distribution, with gamma parameters ``a = df/2``, ``loc = 0`` and
    ``scale = 2``.

    %(after_notes)s

    %(example)s

    c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zchi2_gen._shape_info  r  r2   Nc                 &    |j                  ||      S rH   )	chisquarer  s       r0   r   zchi2_gen._rvs  s    %%b$//r2   c                 L    t        j                  | j                  ||            S rH   r  r  s      r0   rl   zchi2_gen._pdf  s    vvdll1b)**r2   c                     t        j                  |dz  dz
  |      |dz  z
  t        j                  |dz        z
  t        j                  d      |z  dz  z
  S )Nr   r   rO   )rq   rr  r  rJ   r   r  s      r0   r   zchi2_gen._logpdf  sM    xx2a#ad*RZZ2->>"&&)B,PRARRRr2   c                 .    t        j                  ||      S rH   )rq   chdtrr  s      r0   ro   zchi2_gen._cdf      xxAr2   c                 .    t        j                  ||      S rH   )rq   chdtrcr  s      r0   rs   zchi2_gen._sf      yyQr2   c                 .    t        j                  ||      S rH   )rq   chdtrir?   r  r  s      r0   r{   zchi2_gen._isf  r  r2   c                 :    dt        j                  |dz  |      z  S r  rq   r  r  s      r0   rx   zchi2_gen._ppf  s    1a(((r2   c                 \    |}d|z  }dt        j                  d|z        z  }d|z  }||||fS )NrO   r         (@r  r  s         r0   r   zchi2_gen._stats  s=    drwws2v"W3Br2   c                 >    d|z  }d }d }t        |dk  |f||      S )Nr   c                     | t        j                  d      z   t        j                  |       z   d| z
  t        j                  |       z  z   S NrO   r   )rJ   r   rq   r  rT  )half_dfs    r0   r  z*chi2_gen._entropy.<locals>.regular_formula  s>    bffQi'"**W*==[BFF7O34 5r2   c                     t        j                  d      ddt        j                  dt         j                  z        z   z  z   }d| z  }|d|d|d|dz  z   z  z   z  z   z  dt        j                  |       z  z   |z   S )NrO   r   r   gUUUUUUUUUUUUտgllg      @r   )r  r  hs      r0   r  z-chi2_gen._entropy.<locals>.asymptotic_formula  s     q	CRVVAbeeG_!455AGAta51S5=(9!9::;w'(*+, -r2   }   r  r  )r?   r  r  r  r  s        r0   r   zchi2_gen._entropy  s4    (	5		- 'C-')/1 	1r2   r   )r}   r~   r   r   re   r   rl   r   ro   rs   r{   rx   r   r   r   r2   r0   r  r    s<     BF0+S  )1r2   r  r  c                   F    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zy)
cosine_gena\  A cosine continuous random variable.

    %(before_notes)s

    Notes
    -----
    The cosine distribution is an approximation to the normal distribution.
    The probability density function for `cosine` is:

    .. math::

        f(x) = \frac{1}{2\pi} (1+\cos(x))

    for :math:`-\pi \le x \le \pi`.

    %(after_notes)s

    %(example)s

    c                     g S rH   r   rd   s    r0   re   zcosine_gen._shape_infoG  r   r2   c                 Z    dt         j                  z  dt        j                  |      z   z  S Nr   r   rJ   r   r  r   s     r0   rl   zcosine_gen._pdfJ  s!    RUU{AbffQiK((r2   c                 r    t        j                  |      }t        |dk7  |fd t         j                         S )Nr  c                 z    t        j                  |       t        j                  dt         j                  z        z
  S r  )rJ   r  r   r   r  s    r0   r  z$cosine_gen._logpdf.<locals>.<lambda>Q  s#    BHHQK"&&255/$A r2   r  )rJ   r  r   rc   r  s      r0   r   zcosine_gen._logpdfN  s2    FF1I!r'A4A%'VVG- 	-r2   c                 ,    t        j                  |      S rH   rh   _cosine_cdfr   s     r0   ro   zcosine_gen._cdfT  s    q!!r2   c                 .    t        j                  |       S rH   r  r   s     r0   rs   zcosine_gen._sfW  s    r""r2   c                 ,    t        j                  |      S rH   rh   _cosine_invcdfr?   r  s     r0   rx   zcosine_gen._ppfZ  s    !!!$$r2   c                 .    t        j                  |       S rH   r  r  s     r0   r{   zcosine_gen._isf]  s    ""1%%%r2   c                     t         j                  t         j                  z  dz  dz
  }dt         j                  dz  dz
  z  dt         j                  t         j                  z  dz
  dz  z  z  }d	|d	|fS )
Nr;  r   r  r  Z         @r  rO   r   r(  )r?   r  r
  s      r0   r   zcosine_gen._stats`  sa    UURUU]S C'BEE1HrM"cRUURUU]Q->,B&BCAsA~r2   c                 T    t        j                  dt         j                  z        dz
  S )Nr  r   r   rd   s    r0   r   zcosine_gen._entropye  s    vvags""r2   Nr}   r~   r   r   re   rl   r   ro   rs   rx   r{   r   r   r   r2   r0   r  r  2  s4    ()-"#%&
#r2   r  cosinec                   N    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 Zd Zd Zy)
dgamma_gena  A double gamma continuous random variable.

    The double gamma distribution is also known as the reflected gamma
    distribution [1]_.

    %(before_notes)s

    Notes
    -----
    The probability density function for `dgamma` is:

    .. math::

        f(x, a) = \frac{1}{2\Gamma(a)} |x|^{a-1} \exp(-|x|)

    for a real number :math:`x` and :math:`a > 0`. :math:`\Gamma` is the
    gamma function (`scipy.special.gamma`).

    `dgamma` takes ``a`` as a shape parameter for :math:`a`.

    %(after_notes)s

    References
    ----------
    .. [1] Johnson, Kotz, and Balakrishnan, "Continuous Univariate
           Distributions, Volume 1", Second Edition, John Wiley and Sons
           (1994).

    %(example)s

    c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zdgamma_gen._shape_info  r  r2   Nc                     |j                  |      }t        j                  |||      }|t        j                  |dk\  dd      z  S Nr   r  r   r   r  )uniformr  r  rJ   rF  )r?   r   r   r   ugms         r0   r   zdgamma_gen._rvs  sE      d +YYqt,Y?BHHQ#Xq"---r2   c                     t        |      }ddt        j                  |      z  z  ||dz
  z  z  t        j                  |       z  S r  )absrq   r  rJ   r   r?   rk   r   axs       r0   rl   zdgamma_gen._pdf  s>    VAbhhqkM"2#;.<<r2   c                     t        |      }t        j                  |dz
  |      |z
  t        j                  d      z
  t        j
                  |      z
  S r  )r  rq   rr  rJ   r   r  r   s       r0   r   zdgamma_gen._logpdf  s?    VxxC$r)BFF1I5

1EEr2   c           	          t        j                  |dkD  ddt        j                  ||      z  z   dt        j                  ||       z        S Nr   r   )rJ   rF  rq   r  r  r  s      r0   ro   zdgamma_gen._cdf  sF    xxAc"++a"333BLLQB//1 	1r2   c           
          t        j                  |dkD  dt        j                  ||      z  ddt        j                  ||       z  z         S r  )rJ   rF  rq   r  r  r  s      r0   rs   zdgamma_gen._sf  sF    xxABLLA..c"++a!"4446 	6r2   c                 l    t         j                  j                  |      t        j                  d      z
  S Nr   )r  r  r   rJ   r   r  s     r0   r   zdgamma_gen._entropy  s$    {{##A&44r2   c           	          t        j                  |dkD  t        j                  |d|z  dz
        t        j                  |d|z               S r   rJ   rF  rq   r  r  r  s      r0   rx   zdgamma_gen._ppf  sD    xxCq!A#'2AaC002 	2r2   c           	          t        j                  |dkD  t        j                  |d|z  dz
         t        j                  |d|z              S r   r	  r  s      r0   r{   zdgamma_gen._isf  sD    xxC1Q37331Q3/1 	1r2   c                 <    ||dz   z  }d|d|dz   |dz   z  |z  dz
  fS )Nr   r   r   r;  r   )r?   r   r=  s      r0   r   zdgamma_gen._stats  s4    3iCququoc1#555r2   r   )r}   r~   r   r   re   r   rl   r   ro   rs   r   rx   r{   r   r   r2   r0   r  r  l  s;    >E.
=
F1
6
52
1
6r2   r  dgammac                   T    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 Zd Zd Zd Zy)dweibull_genav  A double Weibull continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `dweibull` is given by

    .. math::

        f(x, c) = c / 2 |x|^{c-1} \exp(-|x|^c)

    for a real number :math:`x` and :math:`c > 0`.

    `dweibull` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zdweibull_gen._shape_info  r  r2   Nc                     |j                  |      }t        j                  |||      }|t        j                  |dk\  dd      z  S r  )r  weibull_minr  rJ   rF  )r?   r  r   r   r  ws         r0   r   zdweibull_gen._rvs  sE      d +OOAD|ODBHHQ#Xq"-..r2   c                 l    t        |      }|dz  ||dz
  z  z  t        j                  ||z         z  }|S Nr   r   )r  rJ   r   )r?   rk   r  r  Pxs        r0   rl   zdweibull_gen._pdf  s9    VWrAcE{"RVVRUF^3	r2   c                     t        |      }t        j                  |      t        j                  d      z
  t        j                  |dz
  |      z   ||z  z
  S r  )r  rJ   r   rq   rr  )r?   rk   r  r  s       r0   r   zdweibull_gen._logpdf  sC    Vvvay266#;&!c'2)>>QFFr2   c                     dt        j                  t        |      |z         z  }t        j                  |dkD  d|z
  |      S Nr   r   r   )rJ   r   r  rF  )r?   rk   r  Cx1s       r0   ro   zdweibull_gen._cdf  s:    BFFCFAI:&&xxAq3w,,r2   c                     dt        j                  |dk  |d|z
        z  }t        j                  t        j                  |       d|z        }t        j                  |dkD  ||       S Nr   r   r   )rJ   rF  r  r   )r?   rw   r  r  s       r0   rx   zdweibull_gen._ppf  sX    288AHaa00hhs|S1W-xxCsd++r2   c                     dt         j                  j                  t        j                  |      |      z  }t        j
                  |dkD  |d|z
        S r  )r  r  rs   rJ   r  rF  )r?   rk   r  half_weibull_min_sfs       r0   rs   zdweibull_gen._sf  sF    !E$5$5$9$9"&&)Q$GGxxA2A8K4KLLr2   c                     dt        j                  |dk  |d|z
        z  }t        j                  j	                  ||      }t        j                  |dkD  | |      S r  )rJ   rF  r  r  r{   )r?   rw   r  double_qweibull_min_isfs        r0   r{   zdweibull_gen._isf  sS    c1b1f55++001=xxC/!1?CCr2   c                 P    d|dz  z
  t        j                  dd|z  |z  z         z  S )Nr   rO   r   rq   r  rp  s      r0   r   zdweibull_gen._munp  s+    QUrxxcAgk(9:::r2   c                      yN)r   Nr   Nr   rr  s     r0   r   zdweibull_gen._stats      r2   c                 p    t         j                  j                  |      t        j                  d      z
  }|S r  )r  r  r   rJ   r   )r?   r  r  s      r0   r   zdweibull_gen._entropy  s*    &&q)BFF3K7r2   r   )r}   r~   r   r   re   r   rl   r   ro   rx   rs   r{   r   r   r   r   r2   r0   r  r    sB    *E/
G-,
MD
; r2   r  dweibullc                   ~    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 Zd Zd Zd Ze eed      d               Zy)	expon_genaE  An exponential continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `expon` is:

    .. math::

        f(x) = \exp(-x)

    for :math:`x \ge 0`.

    %(after_notes)s

    A common parameterization for `expon` is in terms of the rate parameter
    ``lambda``, such that ``pdf = lambda * exp(-lambda * x)``. This
    parameterization corresponds to using ``scale = 1 / lambda``.

    The exponential distribution is a special case of the gamma
    distributions, with gamma shape parameter ``a = 1``.

    %(example)s

    c                     g S rH   r   rd   s    r0   re   zexpon_gen._shape_info"  r   r2   Nc                 $    |j                  |      S rH   )standard_exponentialr   s      r0   r   zexpon_gen._rvs%  s    0066r2   c                 .    t        j                  |       S rH   rJ   r   r   s     r0   rl   zexpon_gen._pdf(  s    vvqbzr2   c                     | S rH   r   r   s     r0   r   zexpon_gen._logpdf,  	    r	r2   c                 0    t        j                  |        S rH   rq   r  r   s     r0   ro   zexpon_gen._cdf/      !}r2   c                 0    t        j                  |        S rH   r   r   s     r0   rx   zexpon_gen._ppf2  r3  r2   c                 .    t        j                  |       S rH   r.  r   s     r0   rs   zexpon_gen._sf5  s    vvqbzr2   c                     | S rH   r   r   s     r0   r   zexpon_gen._logsf8  r0  r2   c                 .    t        j                  |       S rH   r+  r   s     r0   r{   zexpon_gen._isf;      q	zr2   c                      y)N)r   r   r         @r   rd   s    r0   r   zexpon_gen._stats>  r   r2   c                      yr  r   rd   s    r0   r   zexpon_gen._entropyA      r2   z        When `method='MLE'`,
        this function uses explicit formulas for the maximum likelihood
        estimation of the exponential distribution parameters, so the
        `optimizer`, `loc` and `scale` keyword arguments are
        ignored.

r   c                    t        |      dkD  rt        d      |j                  dd       }|j                  dd       }t        |       ||t	        d      t        j                  |      }t        j                  |      j                         st	        d      |j                         }||}n#|}||k  rt        d|t
        j                        ||j                         |z
  }n|}t        |      t        |      fS )	Nr   Too many arguments.r   r   r   r   exponr  )r  r.   r-   r1   r   rJ   r   r   r   minrD  rc   r   float)	r?   r@   rA   r/   r   r   data_minr)   r*   s	            r0   r=   zexpon_gen.fitD  s     t9q=122xx%(D)$T* 2 ) * * zz${{4 $$&CDD88:<CC#~"7$bffEE>IIK#%EE Sz5<''r2   r   )r}   r~   r   r   re   r   rl   r   ro   rx   rs   r   r{   r   r   rE   r	   r   r=   r   r2   r0   r)  r)    sf    47"  6 &( &(r2   r)  r?  c                   <    e Zd ZdZd Zd
dZd Zd Zd Zd Z	d	 Z
y)exponnorm_gena  An exponentially modified Normal continuous random variable.

    Also known as the exponentially modified Gaussian distribution [1]_.

    %(before_notes)s

    Notes
    -----
    The probability density function for `exponnorm` is:

    .. math::

        f(x, K) = \frac{1}{2K} \exp\left(\frac{1}{2 K^2} - x / K \right)
                  \text{erfc}\left(-\frac{x - 1/K}{\sqrt{2}}\right)

    where :math:`x` is a real number and :math:`K > 0`.

    It can be thought of as the sum of a standard normal random variable
    and an independent exponentially distributed random variable with rate
    ``1/K``.

    %(after_notes)s

    An alternative parameterization of this distribution (for example, in
    the Wikipedia article [1]_) involves three parameters, :math:`\mu`,
    :math:`\lambda` and :math:`\sigma`.

    In the present parameterization this corresponds to having ``loc`` and
    ``scale`` equal to :math:`\mu` and :math:`\sigma`, respectively, and
    shape parameter :math:`K = 1/(\sigma\lambda)`.

    .. versionadded:: 0.16.0

    References
    ----------
    .. [1] Exponentially modified Gaussian distribution, Wikipedia,
           https://en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution

    %(example)s

    c                 @    t        dddt        j                  fd      gS )NKFr   r  rb   rd   s    r0   re   zexponnorm_gen._shape_info  r  r2   Nc                 V    |j                  |      |z  }|j                  |      }||z   S rH   )r,  r   )r?   rF  r   r   expvalgvals         r0   r   zexponnorm_gen._rvs  s1    22481<++D1}r2   c                 L    t        j                  | j                  ||            S rH   r  )r?   rk   rF  s      r0   rl   zexponnorm_gen._pdf      vvdll1a())r2   c                 p    d|z  }|d|z  |z
  z  }|t        ||z
        z   t        j                  |      z
  S Nr   r   r   rJ   r   )r?   rk   rF  invKexpargs        r0   r   zexponnorm_gen._logpdf  s>    Qwta(QX..::r2   c                     d|z  }|d|z  |z
  z  }|t        ||z
        z   }t        |      t        j                  |      z
  S rM  r   r   rJ   r   r?   rk   rF  rO  rH  logprods         r0   ro   zexponnorm_gen._cdf  sG    Qwta(<D11|bffWo--r2   c                     d|z  }|d|z  |z
  z  }|t        ||z
        z   }t        |       t        j                  |      z   S rM  rR  rS  s         r0   rs   zexponnorm_gen._sf  sI    Qwta(<D11!}rvvg..r2   c                 Z    ||z  }d|z   }d|dz  z  |dz  z  }d|z  |z  |dz  z  }||||fS )Nr   rO   r  r:  r:  r	  r   )r?   rF  K2opK2skwkrts         r0   r   zexponnorm_gen._stats  sO    URx!Q$h%BhmdRj($S  r2   r   )r}   r~   r   r   re   r   rl   r   ro   rs   r   r   r2   r0   rD  rD  w  s,    (RE
*;
./!r2   rD  	exponnormc                 T    t        j                  t        j                  ||             S )a'  
    Compute (1 + x)**y - 1.

    Uses expm1 and xlog1py to avoid loss of precision when
    (1 + x)**y is close to 1.

    Note that the inverse of this function with respect to x is
    ``_pow1pm1(x, 1/y)``.  That is, if

        t = _pow1pm1(x, y)

    then

        x = _pow1pm1(t, 1/y)
    )rJ   r  rq   rq  rk   ys     r0   _pow1pm1r_    s      88BJJq!$%%r2   c                   :    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
y	)
exponweib_gena  An exponentiated Weibull continuous random variable.

    %(before_notes)s

    See Also
    --------
    weibull_min, numpy.random.Generator.weibull

    Notes
    -----
    The probability density function for `exponweib` is:

    .. math::

        f(x, a, c) = a c [1-\exp(-x^c)]^{a-1} \exp(-x^c) x^{c-1}

    and its cumulative distribution function is:

    .. math::

        F(x, a, c) = [1-\exp(-x^c)]^a

    for :math:`x > 0`, :math:`a > 0`, :math:`c > 0`.

    `exponweib` takes :math:`a` and :math:`c` as shape parameters:

    * :math:`a` is the exponentiation parameter,
      with the special case :math:`a=1` corresponding to the
      (non-exponentiated) Weibull distribution `weibull_min`.
    * :math:`c` is the shape parameter of the non-exponentiated Weibull law.

    %(after_notes)s

    References
    ----------
    https://en.wikipedia.org/wiki/Exponentiated_Weibull_distribution

    %(example)s

    c                     t        dddt        j                  fd      }t        dddt        j                  fd      }||gS Nr   Fr   r  r  rb   r?   rb  r  s      r0   re   zexponweib_gen._shape_info  rd  r2   c                 N    t        j                  | j                  |||            S rH   r  r?   rk   r   r  s       r0   rl   zexponweib_gen._pdf	        vvdll1a+,,r2   c                    ||z   }t        j                  |       }t        j                  |      t        j                  |      z   t        j                  |dz
  |      z   |z   t        j                  |dz
  |      z   }|S r  )rq   r  rJ   r   rr  )r?   rk   r   r  negxcexm1clogps          r0   r   zexponweib_gen._logpdf  so    A% q	BFF1I%S%(@@S!,-r2   c                 @    t        j                  ||z          }||z  S rH   r2  )r?   rk   r   r  rj  s        r0   ro   zexponweib_gen._cdf  s!    1a4% axr2   c                 n    t        j                  |d|z  z          t        j                  d|z        z  S r  )rq   r  rJ   r   )r?   rw   r   r  s       r0   rx   zexponweib_gen._ppf  s0    1s1u:+&&CE):::r2   c                 L    t        t        j                  ||z          |       S rH   )r_  rJ   r   rf  s       r0   rs   zexponweib_gen._sf  s"    "&&!Q$-+++r2   c                 X    t        j                  t        | d|z                d|z  z  S rX   )rJ   r   r_  )r?   r  r   r  s       r0   r{   zexponweib_gen._isf  s-    1"ac**++qs33r2   Nr}   r~   r   r   re   rl   r   ro   rx   rs   r{   r   r2   r0   ra  ra    s+    'P
-
;,4r2   ra  	exponweibc                   :    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
y	)
exponpow_gena  An exponential power continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `exponpow` is:

    .. math::

        f(x, b) = b x^{b-1} \exp(1 + x^b - \exp(x^b))

    for :math:`x \ge 0`, :math:`b > 0`.  Note that this is a different
    distribution from the exponential power distribution that is also known
    under the names "generalized normal" or "generalized Gaussian".

    `exponpow` takes ``b`` as a shape parameter for :math:`b`.

    %(after_notes)s

    References
    ----------
    http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Exponentialpower.pdf

    %(example)s

    c                 @    t        dddt        j                  fd      gS Nr   Fr   r  rb   rd   s    r0   re   zexponpow_gen._shape_infoB  r  r2   c                 L    t        j                  | j                  ||            S rH   r  r?   rk   r   s      r0   rl   zexponpow_gen._pdfE      vvdll1a())r2   c                     ||z  }dt        j                  |      z   t        j                  |dz
  |      z   |z   t        j                  |      z
  }|S Nr   r   )rJ   r   rq   rr  r   )r?   rk   r   xbfs        r0   r   zexponpow_gen._logpdfI  sG    Tq	MBHHQWa0025r
Br2   c                 \    t        j                  t        j                  ||z                S rH   r2  rw  s      r0   ro   zexponpow_gen._cdfN  s"    "((1a4.)))r2   c                 Z    t        j                  t        j                  ||z               S rH   rJ   r   rq   r  rw  s      r0   rs   zexponpow_gen._sfQ  s    vvrxx1~o&&r2   c                 `    t        j                  t        j                  |             d|z  z  S r  rq   r  rJ   r   rw  s      r0   r{   zexponpow_gen._isfT  s$    "&&)$1--r2   c                 p    t        t        j                  t        j                  |              d|z        S r  powrq   r  r?   rw   r   s      r0   rx   zexponpow_gen._ppfW  s(    288RXXqb\M*CE22r2   N)r}   r~   r   r   re   rl   r   ro   rs   r{   rx   r   r2   r0   rs  rs  &  s+    6E*
*'.3r2   rs  exponpowc                   `    e Zd ZdZej
                  Zd ZddZd Z	d Z
d Zd Zd	 Zd
 Zd Zy)fatiguelife_gena0  A fatigue-life (Birnbaum-Saunders) continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `fatiguelife` is:

    .. math::

        f(x, c) = \frac{x+1}{2c\sqrt{2\pi x^3}} \exp(-\frac{(x-1)^2}{2x c^2})

    for :math:`x >= 0` and :math:`c > 0`.

    `fatiguelife` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    References
    ----------
    .. [1] "Birnbaum-Saunders distribution",
           https://en.wikipedia.org/wiki/Birnbaum-Saunders_distribution

    %(example)s

    c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zfatiguelife_gen._shape_info{  r  r2   Nc                     |j                  |      }d|z  |z  }||z  }dd|z  z   d|z  t        j                  d|z         z  z   }|S )Nr   r   rO   r   )r   rJ   r   )r?   r  r   r   zrk   x2ts           r0   r   zfatiguelife_gen._rvs~  sT    ((.E!GqS!B$J1RWWQV_,,r2   c                 L    t        j                  | j                  ||            S rH   r  r  s      r0   rl   zfatiguelife_gen._pdf  s     vvdll1a())r2   c                    t        j                  |dz         |dz
  dz  d|z  |dz  z  z  z
  t        j                  d|z        z
  dt        j                  dt         j                  z        dt        j                  |      z  z   z  z
  S )Nr   rO   r   r   r  r   r  s      r0   r   zfatiguelife_gen._logpdf  ss    qsqsQh#a%1*55qsCRVVAbeeG_q{234 	5r2   c                 |    t        d|z  t        j                  |      dt        j                  |      z  z
  z        S r  )r   rJ   r   r  s      r0   ro   zfatiguelife_gen._cdf  s/    qBGGAJRWWQZ$?@AAr2   c                 f    |t        |      z  }d|t        j                  |dz  dz         z   dz  z  S N      ?rO   r  r   rJ   r   r?   rw   r  tmps       r0   rx   zfatiguelife_gen._ppf  s6    )A,sRWWS!VaZ001444r2   c                 |    t        d|z  t        j                  |      dt        j                  |      z  z
  z        S r  )r   rJ   r   r  s      r0   rs   zfatiguelife_gen._sf  s/    a2771:BGGAJ#>?@@r2   c                 h    | t        |      z  }d|t        j                  |dz  dz         z   dz  z  S r  r  r  s       r0   r{   zfatiguelife_gen._isf  s8    b9Q<sRWWS!VaZ001444r2   c                     ||z  }|dz  dz   }d|z  dz   }||z  dz  }d|z  d|z  dz   z  t        j                  |d      z  }d	|z  d
|z  dz   z  |dz  z  }||||fS )Nr   r   r  rA  r     r:  r  r  ]   g      D@rJ   r  )r?   r  c2r<  denr=  r>  r?  s           r0   r   zfatiguelife_gen._stats  s     qS#X^BhnfslUbeck"RXXc3%77Vr"ut|$sCx/3Br2   r   )r}   r~   r   r   r   r  r  re   r   rl   r   ro   rx   rs   r{   r   r   r2   r0   r  r  ^  sD    4 "44ME*
5B5A5r2   r  fatiguelifec                   <    e Zd ZdZd Zd Zd
dZd Zd Zd Z	d	 Z
y)foldcauchy_genao  A folded Cauchy continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `foldcauchy` is:

    .. math::

        f(x, c) = \frac{1}{\pi (1+(x-c)^2)} + \frac{1}{\pi (1+(x+c)^2)}

    for :math:`x \ge 0` and :math:`c \ge 0`.

    `foldcauchy` takes ``c`` as a shape parameter for :math:`c`.

    %(example)s

    c                     |dk\  S Nr   r   rr  s     r0   r]   zfoldcauchy_gen._argcheck      Avr2   c                 @    t        dddt        j                  fd      gS Nr  Fr   ra   rb   rd   s    r0   re   zfoldcauchy_gen._shape_info      3266{MBCCr2   Nc                 D    t        t        j                  |||            S )Nr)   r   r   )r  r  r  r?   r  r   r   s       r0   r   zfoldcauchy_gen._rvs  s&    6::!$+7  9 : 	:r2   c                 d    dt         j                  z  dd||z
  dz  z   z  dd||z   dz  z   z  z   z  S Nr   r   rO   r(  r  s      r0   rl   zfoldcauchy_gen._pdf  s<    255y#q!A#z*S!QqS1H*-==>>r2   c                     dt         j                  z  t        j                  ||z
        t        j                  ||z         z   z  S r  r|  r  s      r0   ro   zfoldcauchy_gen._cdf  s2    255y"))AaC.299QqS>9::r2   c                     t        j                  d||z
        t        j                  d||z         z   t         j                  z  S rX   )rJ   arctan2r   r  s      r0   rs   zfoldcauchy_gen._sf  s6    
 

1a!e$rzz!QU';;RUUBBr2   c                 ~    t         j                  t         j                  t         j                  t         j                  fS rH   r  rr  s     r0   r   zfoldcauchy_gen._stats  r  r2   r   r}   r~   r   r   r]   re   r   rl   ro   rs   r   r   r2   r0   r  r    s,    &D:?;C.r2   r  
foldcauchyc                   H    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 Zd Zy)f_gena  An F continuous random variable.

    For the noncentral F distribution, see `ncf`.

    %(before_notes)s

    See Also
    --------
    ncf

    Notes
    -----
    The F distribution with :math:`df_1 > 0` and :math:`df_2 > 0` degrees of freedom is
    the distribution of the ratio of two independent chi-squared distributions with
    :math:`df_1` and :math:`df_2` degrees of freedom, after rescaling by
    :math:`df_2 / df_1`.
    
    The probability density function for `f` is:

    .. math::

        f(x, df_1, df_2) = \frac{df_2^{df_2/2} df_1^{df_1/2} x^{df_1 / 2-1}}
                                {(df_2+df_1 x)^{(df_1+df_2)/2}
                                 B(df_1/2, df_2/2)}

    for :math:`x > 0`.

    `f` accepts shape parameters ``dfn`` and ``dfd`` for :math:`df_1`, the degrees of
    freedom of the chi-squared distribution in the numerator, and :math:`df_2`, the
    degrees of freedom of the chi-squared distribution in the denominator, respectively.

    %(after_notes)s

    %(example)s

    c                     t        dddt        j                  fd      }t        dddt        j                  fd      }||gS )NdfnFr   r  dfdrb   )r?   idfnidfds      r0   re   zf_gen._shape_info	  s<    %BFF^D%BFF^Dd|r2   Nc                 (    |j                  |||      S rH   )r|  )r?   r  r  r   r   s        r0   r   z
f_gen._rvs	  s    ~~c3--r2   c                 N    t        j                  | j                  |||            S rH   r  r?   rk   r  r  s       r0   rl   z
f_gen._pdf	  s      vvdll1c3/00r2   c                 F   d|z  }d|z  }|dz  t        j                  |      z  |dz  t        j                  |      z  z   t        j                  |dz  dz
  |      z   ||z   dz  t        j                  |||z  z         z  t        j                  |dz  |dz        z   z
  }|S Nr   rO   r   )rJ   r   rq   rr  rs  )r?   rk   r  r  r\   mrt  s          r0   r   zf_gen._logpdf	  s    #I#IsRVVAY1rvvay0288AaC!GQ3GGaC7bffQ1Wo-		!A#qs0CCE
r2   c                 0    t        j                  |||      S rH   )rq   fdtrr  s       r0   ro   z
f_gen._cdf	  s    wwsC##r2   c                 0    t        j                  |||      S rH   )rq   fdtrcr  s       r0   rs   z	f_gen._sf	      xxS!$$r2   c                 0    t        j                  |||      S rH   )rq   fdtri)r?   rw   r  r  s       r0   rx   z
f_gen._ppf!	  r  r2   c                    d|z  d|z  }}|dz
  |dz
  |dz
  |dz
  f\  }}}}t        |dkD  ||fd t        j                        }	t        |dkD  ||||fd	 t        j                        }
t        |d
kD  ||||fd t        j                        }|t        j                  d      z  }t        |dkD  |||fd t        j                        }|dz  }|	|
||fS )Nr   r   rA  r:         @rO   c                     | |z  S rH   r   )v2v2_2s     r0   r  zf_gen._stats.<locals>.<lambda>*	  s
    R$Y r2   r  c                 6    d|z  |z  | |z   z  | |dz  z  |z  z  S r  r   )v1r  r  v2_4s       r0   r  zf_gen._stats.<locals>.<lambda>/	  s-    FRK29%dAg)<= r2   r  c                 V    d| z  |z   |z  t        j                  || | |z   z  z        z  S r  r  )r  r  r  v2_6s       r0   r  zf_gen._stats.<locals>.<lambda>5	  s3    Vd]d"RWWTR295E-F%GG r2   r'  c                     d| | z  |z  z   |z  S )Nr'  r   )r>  r  v2_8s      r0   r  zf_gen._stats.<locals>.<lambda><	  s    AR$$6$#> r2   r  )r   rJ   rc   r  r   )r?   r  r  r  r  r  r  r  r  r<  r=  r>  r?  s                r0   r   zf_gen._stats$	  s   c28B!#b"r'27BG!CdD$FRJ&FF
 FRT4(>FF	 FRtT*HFF	
 	bggbkFRt$>FF 	g3Br2   c                 L   d|z  }d|z  }d||z   z  }t        j                  |      t        j                  |      z
  t        j                  ||      z   d|z
  t        j                  |      z  z   d|z   t        j                  |      z  z
  |t        j                  |      z  z   S r  )rJ   r   rq   rs  rT  )r?   r  r  half_dfnhalf_dfdhalf_sums         r0   r   zf_gen._entropyB	  s     99#)$sbffSk)BIIh,IIX!11256\x 5!!#+bffX.>#>? 	@r2   r   )r}   r~   r   r   re   r   rl   r   ro   rs   rx   r   r   r   r2   r0   r  r    s6    #H
.1$%%<
@r2   r  r|  c                   <    e Zd ZdZd Zd Zd
dZd Zd Zd Z	d	 Z
y)foldnorm_genaz  A folded normal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `foldnorm` is:

    .. math::

        f(x, c) = \sqrt{2/\pi} cosh(c x) \exp(-\frac{x^2+c^2}{2})

    for :math:`x \ge 0` and :math:`c \ge 0`.

    `foldnorm` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                     |dk\  S r  r   rr  s     r0   r]   zfoldnorm_gen._argcheckp	  r  r2   c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zfoldnorm_gen._shape_infos	  r  r2   Nc                 <    t        |j                  |      |z         S rH   r  r   r  s       r0   r   zfoldnorm_gen._rvsv	  s    <//59::r2   c                 <    t        ||z         t        ||z
        z   S rH   r   r  s      r0   rl   zfoldnorm_gen._pdfy	  s    Q)AaC.00r2   c                     t        j                  d      }dt        j                  ||z
  |z        t        j                  ||z   |z        z   z  S r  )rJ   r   rq   erf)r?   rk   r  sqrt_twos       r0   ro   zfoldnorm_gen._cdf}	  sC    771:bffa!eX-.Q8H1IIJJr2   c                 <    t        ||z
        t        ||z         z   S rH   r   r  s      r0   rs   zfoldnorm_gen._sf	  s    A!a%00r2   c                    ||z  }t        j                  d|z        t        j                  dt         j                  z        z  }d|z  |t	        j
                  |t        j                  d      z        z  z   }|dz   ||z  z
  }d||z  |z  ||z  z
  |z
  z  }|t        j                  |d      z  }||dz   z  dz   d|z  |z  z   }|d|d	z
  z  d	|dz  z  z
  |dz  z  z  }||dz  z  d	z
  }||||fS )
N      r   rO   r   r  r:  r  r  r;  )rJ   r   r   r   rq   r  r  )r?   r  r  expfacr<  r=  r>  r?  s           r0   r   zfoldnorm_gen._stats	  s	    qSR2772bee8#44YRVVAbggajL1111fr"un2b58be#f,-
bhhsC  27^a"V)B,.
rR"W~RU
*b!e33#s(]R3Br2   r   r  r   r2   r0   r  r  Z	  s,    *D;1K1r2   r  foldnormc                   x     e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zd Z eed       fd       Z xZS )weibull_min_gena  Weibull minimum continuous random variable.

    The Weibull Minimum Extreme Value distribution, from extreme value theory
    (Fisher-Gnedenko theorem), is also often simply called the Weibull
    distribution. It arises as the limiting distribution of the rescaled
    minimum of iid random variables.

    %(before_notes)s

    See Also
    --------
    weibull_max, numpy.random.Generator.weibull, exponweib

    Notes
    -----
    The probability density function for `weibull_min` is:

    .. math::

        f(x, c) = c x^{c-1} \exp(-x^c)

    for :math:`x > 0`, :math:`c > 0`.

    `weibull_min` takes ``c`` as a shape parameter for :math:`c`.
    (named :math:`k` in Wikipedia article and :math:`a` in
    ``numpy.random.weibull``).  Special shape values are :math:`c=1` and
    :math:`c=2` where Weibull distribution reduces to the `expon` and
    `rayleigh` distributions respectively.

    Suppose ``X`` is an exponentially distributed random variable with
    scale ``s``. Then ``Y = X**k`` is `weibull_min` distributed with shape
    ``c = 1/k`` and scale ``s**k``.

    %(after_notes)s

    References
    ----------
    https://en.wikipedia.org/wiki/Weibull_distribution

    https://en.wikipedia.org/wiki/Fisher-Tippett-Gnedenko_theorem

    %(example)s

    c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zweibull_min_gen._shape_info	  r  r2   c                 h    |t        ||dz
        z  t        j                  t        ||             z  S rX   r  rJ   r   r  s      r0   rl   zweibull_min_gen._pdf	  s,    Q!}RVVSAYJ///r2   c                 z    t        j                  |      t        j                  |dz
  |      z   t	        ||      z
  S rX   rJ   r   rq   rr  r  r  s      r0   r   zweibull_min_gen._logpdf	  s/    vvay288AE1--Aq	99r2   c                 D    t        j                  t        ||              S rH   rq   r  r  r  s      r0   ro   zweibull_min_gen._cdf	  s    #a)$$$r2   c                 J    t        t        j                  |        d|z        S r  r  r  s      r0   rx   zweibull_min_gen._ppf	  s    BHHaRL=#a%((r2   c                 L    t        j                  | j                  ||            S rH   r-  r  s      r0   rs   zweibull_min_gen._sf	      vvdkk!Q'((r2   c                     t        ||       S rH   r  r  s      r0   r   zweibull_min_gen._logsf	  s    Aq	zr2   c                 :    t        j                  |       d|z  z  S rX   r+  r  s      r0   r{   zweibull_min_gen._isf	  s    
ac""r2   c                 >    t        j                  d|dz  |z  z         S r  r"  rp  s      r0   r   zweibull_min_gen._munp	  s    xxAcE!G$$r2   c                 V    t          |z  t        j                  |      z
  t         z   dz   S rX   r    rJ   r   rr  s     r0   r   zweibull_min_gen._entropy	  %    w{RVVAY&/!33r2   a          If ``method='mm'``, parameters fixed by the user are respected, and the
        remaining parameters are used to match distribution and sample moments
        where possible. For example, if the user fixes the location with
        ``floc``, the parameters will only match the distribution skewness and
        variance to the sample skewness and variance; no attempt will be made
        to match the means or minimize a norm of the errors.
        

r   c           	      4   t        |t              r7|j                         dk(  r|j                         }nt	        |   |g|i |S |j                  dd      rt	        |   |g|i |S t        | |||      \  }}}}|j                  dd      j                         }d t        j                  |      d} |      }	|	k  r|dk7  r||st	        |   |g|i |S |dk(  rd	\  }
}}n6t        |      r|d   nd }
|j                  d
d       }|j                  dd       }|!|
t        fdd|gd      j                  }
n||}
|h|ft        j                   |      }t        j"                  |t%        j&                  dd|
z  z         t%        j&                  dd|
z  z         dz  z
  z        }n||}|9|7t        j(                  |      }||t%        j&                  dd|
z  z         z  z
  }n||}|dk(  r|
||fS t	        |   ||
f||d|S )Nr   superfitFr,   r5   c                     t        j                  dd| z  z         }t        j                  dd| z  z         }t        j                  dd| z  z         }d|dz  z  d|z  |z  z
  |z   }||dz  z
  dz  }||z  S )Nr   rO   r  r  r"  )r  gamma1gamma2gamma3numr  s         r0   skewz!weibull_min_gen.fit.<locals>.skew
  s|    XXa!e_FXXa!e_FXXa!e_Ffai-!F(6/1F:CFAI%-Cs7Nr2   g     @r6   NNNr)   r*   c                      |       z
  S rH   r   )r  r  r   s    r0   r  z%weibull_min_gen.fit.<locals>.<lambda>#
  s    d1gk r2   g{Gz?bisect)bracketr,   r   rO   r)   r*   )r9   r%   r:   r  r;   r=   r-   _check_fit_input_parametersr7   r8   r  r   r  r&   rootrJ   r  r   rq   r  r   )r?   r@   rA   r/   fcr   r   r,   max_cs_minr  r)   r*   r  r  r  r   r  s                  @@r0   r=   zweibull_min_gen.fit	  s<    dL)  "a'~~'w{47$7$7788J&7;t3d3d33 "=T4=A4"Ib$(E*002	 JJtUu94BJt7;t3d3d33 T>,MAsEt9Q$A((5$'CHHWd+E:!) 1D%=#+--1T ^A>emtAGGA!AaC%288AacE?A3E!EFGEE<CKAeBHHQ1W---CCT>c5=  7;tQECuEEEr2   )r}   r~   r   r   re   rl   r   ro   rx   rs   r   r{   r   r   r   r   r=   r  r  s   @r0   r  r  	  s`    +XE0:%))#%4 } 5 JFJFr2   r  r  c                   j     e Zd ZdZd Zd Z fdZd Zd Zd Z	d Z
d	 Zd
 Zd Zd Zd Zd Z xZS )truncweibull_min_gena9  A doubly truncated Weibull minimum continuous random variable.

    %(before_notes)s

    See Also
    --------
    weibull_min, truncexpon

    Notes
    -----
    The probability density function for `truncweibull_min` is:

    .. math::

        f(x, a, b, c) = \frac{c x^{c-1} \exp(-x^c)}{\exp(-a^c) - \exp(-b^c)}

    for :math:`a < x <= b`, :math:`0 \le a < b` and :math:`c > 0`.

    `truncweibull_min` takes :math:`a`, :math:`b`, and :math:`c` as shape
    parameters.

    Notice that the truncation values, :math:`a` and :math:`b`, are defined in
    standardized form:

    .. math::

        a = (u_l - loc)/scale
        b = (u_r - loc)/scale

    where :math:`u_l` and :math:`u_r` are the specific left and right
    truncation values, respectively. In other words, the support of the
    distribution becomes :math:`(a*scale + loc) < x <= (b*scale + loc)` when
    :math:`loc` and/or :math:`scale` are provided.

    %(after_notes)s

    References
    ----------

    .. [1] Rinne, H. "The Weibull Distribution: A Handbook". CRC Press (2009).

    %(example)s

    c                 $    |dk\  ||kD  z  |dkD  z  S Nr   r   r?   r  r   r   s       r0   r]   ztruncweibull_min_gen._argcheckl
  s    RAE"a"f--r2   c                     t        dddt        j                  fd      }t        dddt        j                  fd      }t        dddt        j                  fd      }|||gS )Nr  Fr   r  r   ra   r   rb   )r?   r  rb  rc  s       r0   re   z truncweibull_min_gen._shape_infoo
  sV    UQK@UQK?UQK@B|r2   c                 &    t         |   |d      S )N)r   r   r   rF  r;   r  r?   r@   r  s     r0   r  ztruncweibull_min_gen._fitstartu
  s    w I 66r2   c                 
    ||fS rH   r   r  s       r0   r   z!truncweibull_min_gen._get_supporty
      !tr2   c                     t        j                  t        ||             t        j                  t        ||             z
  }|t        ||dz
        z  t        j                  t        ||             z  |z  S rX   rJ   r   r  )r?   rk   r  r   r   denums         r0   rl   ztruncweibull_min_gen._pdf|
  s]    Q
#bffc!QiZ&88C1Q3K"&&#a)"44==r2   c           	      (   t        j                  t        j                  t        ||             t        j                  t        ||             z
        }t        j                  |      t	        j
                  |dz
  |      z   t        ||      z
  |z
  S rX   )rJ   r   r   r  rq   rr  )r?   rk   r  r   r   logdenums         r0   r   ztruncweibull_min_gen._logpdf
  si    66"&&#a),rvvs1ayj/AABvvay288AE1--Aq	9HDDr2   c                    t        j                  t        ||             t        j                  t        ||             z
  }t        j                  t        ||             t        j                  t        ||             z
  }||z  S rH   r  r?   rk   r  r   r   r  r  s          r0   ro   ztruncweibull_min_gen._cdf
  d    vvs1ayj!BFFC1I:$66Q
#bffc!QiZ&88U{r2   c           	      \   t        j                  t        j                  t        ||             t        j                  t        ||             z
        }t        j                  t        j                  t        ||             t        j                  t        ||             z
        }||z
  S rH   rJ   r   r   r  r?   rk   r  r   r   lognumr  s          r0   r   ztruncweibull_min_gen._logcdf
  w    Aq	z*RVVSAYJ-??@66"&&#a),rvvs1ayj/AAB  r2   c                    t        j                  t        ||             t        j                  t        ||             z
  }t        j                  t        ||             t        j                  t        ||             z
  }||z  S rH   r  r  s          r0   rs   ztruncweibull_min_gen._sf
  r  r2   c           	      \   t        j                  t        j                  t        ||             t        j                  t        ||             z
        }t        j                  t        j                  t        ||             t        j                  t        ||             z
        }||z
  S rH   r  r   s          r0   r   ztruncweibull_min_gen._logsf
  r"  r2   c                     t        t        j                  d|z
  t        j                  t        ||             z  |t        j                  t        ||             z  z          d|z        S rX   r  rJ   r   r   r?   rw   r  r   r   s        r0   r{   ztruncweibull_min_gen._isf
  Y    VVQUbffc!QiZ001rvvs1ayj7I3IIJJAaC 	r2   c                     t        t        j                  d|z
  t        j                  t        ||             z  |t        j                  t        ||             z  z          d|z        S rX   r&  r'  s        r0   rx   ztruncweibull_min_gen._ppf
  r(  r2   c           	      `   t        j                  ||z  dz         t        j                  ||z  dz   t        ||            t        j                  ||z  dz   t        ||            z
  z  }t	        j
                  t        ||             t	        j
                  t        ||             z
  }||z  S r  )rq   r  r  r  rJ   r   )r?   r\   r  r   r   	gamma_funr  s          r0   r   ztruncweibull_min_gen._munp
  s    HHQqS2X&KK!b#a),r{{1Q38SAY/OO	 Q
#bffc!QiZ&885  r2   )r}   r~   r   r   r]   re   r  r   rl   r   ro   r   rs   r   r{   rx   r   r  r  s   @r0   r  r  ?
  sK    +X.7>E
!

!


!r2   r  truncweibull_minc                   F    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zy)weibull_max_gena0  Weibull maximum continuous random variable.

    The Weibull Maximum Extreme Value distribution, from extreme value theory
    (Fisher-Gnedenko theorem), is the limiting distribution of rescaled
    maximum of iid random variables. This is the distribution of -X
    if X is from the `weibull_min` function.

    %(before_notes)s

    See Also
    --------
    weibull_min

    Notes
    -----
    The probability density function for `weibull_max` is:

    .. math::

        f(x, c) = c (-x)^{c-1} \exp(-(-x)^c)

    for :math:`x < 0`, :math:`c > 0`.

    `weibull_max` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    References
    ----------
    https://en.wikipedia.org/wiki/Weibull_distribution

    https://en.wikipedia.org/wiki/Fisher-Tippett-Gnedenko_theorem

    %(example)s

    c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zweibull_max_gen._shape_info
  r  r2   c                 l    |t        | |dz
        z  t        j                  t        | |             z  S rX   r  r  s      r0   rl   zweibull_max_gen._pdf
  s0    aR1~bffc1"aj[111r2   c                 ~    t        j                  |      t        j                  |dz
  |       z   t	        | |      z
  S rX   r  r  s      r0   r   zweibull_max_gen._logpdf
  s3    vvay288AaC!,,sA2qz99r2   c                 D    t        j                  t        | |             S rH   r  r  s      r0   ro   zweibull_max_gen._cdf
  s    vvsA2qzk""r2   c                     t        | |       S rH   r  r  s      r0   r   zweibull_max_gen._logcdf
  s    QB
{r2   c                 F    t        j                  t        | |              S rH   r  r  s      r0   rs   zweibull_max_gen._sf
  s    #qb!*%%%r2   c                 J    t        t        j                  |       d|z         S r  )r  rJ   r   r  s      r0   rx   zweibull_max_gen._ppf
  s     RVVAYJA&&&r2   c                 v    t        j                  d|dz  |z  z         }t        |      dz  rd}||z  S d}||z  S )Nr   rO   r  r   )rq   r  r  )r?   r\   r  valsgns        r0   r   zweibull_max_gen._munp
  sI    hhs1S57{#q6A:C Sy CSyr2   c                 V    t          |z  t        j                  |      z
  t         z   dz   S rX   r  rr  s     r0   r   zweibull_max_gen._entropy
  r  r2   N)r}   r~   r   r   re   rl   r   ro   r   rs   rx   r   r   r   r2   r0   r.  r.  
  s6    #HE2:#&'4r2   r.  weibull_max)r   r   c                   L    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zd Zy)genlogistic_gena  A generalized logistic continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `genlogistic` is:

    .. math::

        f(x, c) = c \frac{\exp(-x)}
                         {(1 + \exp(-x))^{c+1}}

    for real :math:`x` and :math:`c > 0`. In literature, different
    generalizations of the logistic distribution can be found. This is the type 1
    generalized logistic distribution according to [1]_. It is also referred to
    as the skew-logistic distribution [2]_.

    `genlogistic` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    References
    ----------
    .. [1] Johnson et al. "Continuous Univariate Distributions", Volume 2,
           Wiley. 1995.
    .. [2] "Generalized Logistic Distribution", Wikipedia,
           https://en.wikipedia.org/wiki/Generalized_logistic_distribution

    %(example)s

    c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zgenlogistic_gen._shape_info  r  r2   c                 L    t        j                  | j                  ||            S rH   r  r  s      r0   rl   zgenlogistic_gen._pdf  rx  r2   c                     |dz
   |dk  z  dz
  }t        j                  |      }t        j                  |      ||z  z   |dz   t        j                  t        j
                  |             z  z
  S Nr   r   )rJ   r  r   rq   r  r   )r?   rk   r  multabsxs        r0   r   zgenlogistic_gen._logpdf  sb     Qx1q5!A%vvayvvay49$!rxxu/F'FFFr2   c                 @    dt        j                  |       z   | z  }|S rX   r.  )r?   rk   r  Cxs       r0   ro   zgenlogistic_gen._cdf'  s!    r
lqb!	r2   c                 \    | t        j                  t        j                  |             z  S rH   )rJ   r  r   r  s      r0   r   zgenlogistic_gen._logcdf+  s"    rBHHRVVQBZ(((r2   c                 \    t        j                  t        j                  |d|z               S r6  )rJ   r   rq   powm1r  s      r0   rx   zgenlogistic_gen._ppf.  s#    rxx46*+++r2   c                 N    t        j                  | j                  ||             S rH   rq   r  r   r  s      r0   rs   zgenlogistic_gen._sf1      a+,,,r2   c                 ,    | j                  d|z
  |      S rX   rx   r  s      r0   r{   zgenlogistic_gen._isf4  s    yyQ""r2   c                    t         t        j                  |      z   }t        j                  t        j                  z  dz  t        j
                  d|      z   }dt        j
                  d|      z  dt        z  z   }|t        j                  |d      z  }t        j                  dz  dz  dt        j
                  d|      z  z   }||d	z  z  }||||fS )
Nr:  rO   r	  r  r  r        .@r  r   )r    rq   rT  rJ   r   zetar!   r  r?   r  r<  r=  r>  r?  s         r0   r   zgenlogistic_gen._stats7  s    bffQieeBEEk#o1-1&(
bhhsC  UUAXd]Qrwwq!}_,
c3h3Br2   c                 ,    t        |dk  |fd d       S )Ng    ^Ac                 t    t        j                  |        t        j                  | dz         z   t        z   dz   S rX   )rJ   r   rq   rT  r    r  s    r0   r  z*genlogistic_gen._entropy.<locals>.<lambda>B  s+    RVVAYJA$>$G!$K r2   c                 &    dd| z  z  t         z   dz   S r*  r    r  s    r0   r  z*genlogistic_gen._entropy.<locals>.<lambda>H  s    q!a%y6'9A'= r2   r  r  rr  s     r0   r   zgenlogistic_gen._entropy@  s!    !c'A5K >? 	?r2   N)r}   r~   r   r   re   rl   r   ro   r   rx   rs   r{   r   r   r   r2   r0   r<  r<  
  s<    @E*G),-#?r2   r<  genlogisticc                   `    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zd ZddZd Zd Zy)genpareto_gena  A generalized Pareto continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `genpareto` is:

    .. math::

        f(x, c) = (1 + c x)^{-1 - 1/c}

    defined for :math:`x \ge 0` if :math:`c \ge 0`, and for
    :math:`0 \le x \le -1/c` if :math:`c < 0`.

    `genpareto` takes ``c`` as a shape parameter for :math:`c`.

    For :math:`c=0`, `genpareto` reduces to the exponential
    distribution, `expon`:

    .. math::

        f(x, 0) = \exp(-x)

    For :math:`c=-1`, `genpareto` is uniform on ``[0, 1]``:

    .. math::

        f(x, -1) = 1

    %(after_notes)s

    %(example)s

    c                 ,    t        j                  |      S rH   rJ   r   rr  s     r0   r]   zgenpareto_gen._argcheckr      {{1~r2   c                 ^    t        ddt        j                   t        j                  fd      gS Nr  Fr  rb   rd   s    r0   re   zgenpareto_gen._shape_infou  %    3'8.IJJr2   c                     t        j                  |      }t        |dk  |fd t         j                        }t        j                  |dk\  | j
                  | j
                        }||fS )Nr   c                     d| z  S r6  r   r  s    r0   r  z,genpareto_gen._get_support.<locals>.<lambda>{  s
    q r2   )rJ   r   r   rc   rF  r   )r?   r  r   r   s       r0   r   zgenpareto_gen._get_supportx  sV    JJqMq1uqd(vv HHQ!VTVVTVV,!tr2   c                 L    t        j                  | j                  ||            S rH   r  r  s      r0   rl   zgenpareto_gen._pdf  rx  r2   c                 8    t        ||k(  |dk7  z  ||fd |       S )Nr   c                 B    t        j                  |dz   || z         |z  S r  r]  rk   r  s     r0   r  z'genpareto_gen._logpdf.<locals>.<lambda>  s!    

1r61Q3(?'?!'C r2   r  r  s      r0   r   zgenpareto_gen._logpdf  s,    16a1f-1vC" 	r2   c                 4    t        j                  | |        S rH   )rq   inv_boxcox1pr  s      r0   ro   zgenpareto_gen._cdf  s    QB'''r2   c                 2    t        j                  | |       S rH   )rq   
inv_boxcoxr  s      r0   rs   zgenpareto_gen._sf  s    }}aR!$$r2   c                 8    t        ||k(  |dk7  z  ||fd |       S )Nr   c                 :    t        j                  || z         |z  S rH   r   rc  s     r0   r  z&genpareto_gen._logsf.<locals>.<lambda>  s    1~'9 r2   r  r  s      r0   r   zgenpareto_gen._logsf  s,    16a1f-1v9" 	r2   c                 4    t        j                  | |        S rH   )rq   boxcox1pr  s      r0   rx   zgenpareto_gen._ppf  s    QB###r2   c                 2    t        j                  ||        S rH   )rq   boxcoxr  s      r0   r{   zgenpareto_gen._isf  s    		!aR   r2   c                 N   d|vrd }n!t        |dk  |fd t        j                        }d|vrd }n!t        |dk  |fd t        j                        }d|vrd }n!t        |dk  |fd	 t        j                        }d
|vrd }n!t        |dk  |fd t        j                        }||||fS )Nr  r   c                     dd| z
  z  S rX   r   xis    r0   r  z&genpareto_gen._stats.<locals>.<lambda>  s    aRj r2   r  r   c                 *    dd| z
  dz  z  dd| z  z
  z  S r*  r   rp  s    r0   r  z&genpareto_gen._stats.<locals>.<lambda>  s    a1r6A+oQrT&B r2   r  gUUUUUU?c                 \    dd| z   z  t        j                  dd| z  z
        z  dd| z  z
  z  S )NrO   r   r  r  rp  s    r0   r  z&genpareto_gen._stats.<locals>.<lambda>  s4    qAF|bgga!B$h6G'G()AbD(2 r2   r
  r  c                 `    ddd| z  z
  z  d| dz  z  | z   dz   z  dd| z  z
  z  dd| z  z
  z  dz
  S )Nr  r   rO   r  r   rp  s    r0   r  z&genpareto_gen._stats.<locals>.<lambda>  sQ    qA"H~2q529I'J()AbD(2562X(?AB(C r2   r   rJ   rc   r  )r?   r  r  r  r  r  r
  s          r0   r   zgenpareto_gen._stats  s    gA1q51$066#A gA1s7QDB66#A gA1s7QD366#A gA1s7QDD66#A !Qzr2   c           	      d    d t        |dk7  |ffdt        j                  dz               S )Nc                    d}t        j                  d| dz         }t        |t        j                  | |            D ]  \  }}||d|z  z  d||z  z
  z  z   } t        j
                  || z  dk  |d|z  | z  z  t         j                        S )Nr   r   r   r  r   r3  )rJ   rD  ziprq   combrF  rc   )r\   r  r7  r
  kicnks         r0   rM  z#genpareto_gen._munp.<locals>.__munp  s    C		!QU#Aq"''!Q-0 >CC2"*,a"f==>88AEAIsdQh1_'<bffEEr2   r   c                      |       S rH   r   )r  _genpareto_gen__munpr\   s    r0   r  z%genpareto_gen._munp.<locals>.<lambda>  s    F1aL r2   r   )r   rq   r  )r?   r\   r  r}  s    ` @r0   r   zgenpareto_gen._munp  s4    	F !q&1$0((1q5/+ 	+r2   c                     d|z   S r  r   rr  s     r0   r   zgenpareto_gen._entropy      Avr2   Nr  )r}   r~   r   r   r]   re   r   rl   r   ro   rs   r   rx   r{   r   r   r   r   r2   r0   rW  rW  N  sJ    "FK*
(%
$!:	+r2   rW  	genparetoc                   :    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
y	)
genexpon_gena!  A generalized exponential continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `genexpon` is:

    .. math::

        f(x, a, b, c) = (a + b (1 - \exp(-c x)))
                        \exp(-a x - b x + \frac{b}{c}  (1-\exp(-c x)))

    for :math:`x \ge 0`, :math:`a, b, c > 0`.

    `genexpon` takes :math:`a`, :math:`b` and :math:`c` as shape parameters.

    %(after_notes)s

    References
    ----------
    H.K. Ryu, "An Extension of Marshall and Olkin's Bivariate Exponential
    Distribution", Journal of the American Statistical Association, 1993.

    N. Balakrishnan, Asit P. Basu (editors), *The Exponential Distribution:
    Theory, Methods and Applications*, Gordon and Breach, 1995.
    ISBN 10: 2884491929

    %(example)s

    c                     t        dddt        j                  fd      }t        dddt        j                  fd      }t        dddt        j                  fd      }|||gS )Nr   Fr   r  r   r  rb   )r?   rb  rc  r  s       r0   re   zgenexpon_gen._shape_info  sV    UQK@UQK@UQK@B|r2   c           	          ||t        j                  | |z         z  z   t        j                  | |z
  |z  |t        j                  | |z         z  |z  z         z  S rH   rq   r  rJ   r   r?   rk   r   r   r  s        r0   rl   zgenexpon_gen._pdf  sh     A!A''!Aq01BHHaRTN?0CA0E1F *G G 	Gr2   c                     t        j                  ||t        j                  | |z         z  z         | |z
  |z  z   |t        j                  | |z         z  |z  z   S rH   rJ   r   rq   r  r  s        r0   r   zgenexpon_gen._logpdf  sY    vvaBHHaRTN?++,1ax7BHHaRTN?8KA8MMMr2   c                 ~    t        j                  | |z
  |z  |t        j                  | |z         z  |z  z          S rH   r2  r  s        r0   ro   zgenexpon_gen._cdf  s=    1"Q$A!A$7$99:::r2   c                     ||z   }||t        j                  |       z  z
  |z  }|t        j                  | |z  t        j                  |       z        j
                  z   |z  S rH   )rJ   r  rq   lambertwr   realr?   r  r   r   r  r  r  s          r0   rx   zgenexpon_gen._ppf  s\    E288QB<"BKK1rvvqbz 12777::r2   c                 |    t        j                  | |z
  |z  |t        j                  | |z         z  |z  z         S rH   r  r  s        r0   rs   zgenexpon_gen._sf   s:    vvr!tQhRXXqbd^O!4Q!6677r2   c                     ||z   }||t        j                  |      z  z
  |z  }|t        j                  | |z  t        j                  |       z        j
                  z   |z  S rH   )rJ   r   rq   r  r   r  r  s          r0   r{   zgenexpon_gen._isf  sY    E266!9_aBKK1rvvqbz 12777::r2   Nrp  r   r2   r0   r  r    s,    >GN;;
8;r2   r  genexponc                   v     e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zd Zd Zd Z fdZd Zd Z xZS )genextreme_genaB  A generalized extreme value continuous random variable.

    %(before_notes)s

    See Also
    --------
    gumbel_r

    Notes
    -----
    For :math:`c=0`, `genextreme` is equal to `gumbel_r` with
    probability density function

    .. math::

        f(x) = \exp(-\exp(-x)) \exp(-x),

    where :math:`-\infty < x < \infty`.

    For :math:`c \ne 0`, the probability density function for `genextreme` is:

    .. math::

        f(x, c) = \exp(-(1-c x)^{1/c}) (1-c x)^{1/c-1},

    where :math:`-\infty < x \le 1/c` if :math:`c > 0` and
    :math:`1/c \le x < \infty` if :math:`c < 0`.

    Note that several sources and software packages use the opposite
    convention for the sign of the shape parameter :math:`c`.

    `genextreme` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                 ,    t        j                  |      S rH   rY  rr  s     r0   r]   zgenextreme_gen._argcheck3  rZ  r2   c                 ^    t        ddt        j                   t        j                  fd      gS r\  rb   rd   s    r0   re   zgenextreme_gen._shape_info6  r]  r2   c                    t        j                  |dkD  dt        j                  |t              z  t         j                        }t        j                  |dk  dt        j
                  |t               z  t         j                         }||fS Nr   r   )rJ   rF  maximumr   rc   minimum)r?   r  _b_as       r0   r   zgenextreme_gen._get_support9  sc    XXa!eS2::a#77@XXa!eS2::a%#88266'B2vr2   c                 8    t        ||k(  |dk7  z  ||fd |       S )Nr   c                 :    t        j                  | | z        |z  S rH   r   rc  s     r0   r  z+genextreme_gen._loglogcdf.<locals>.<lambda>A  s    rxx1~a'7 r2   r  r  s      r0   
_loglogcdfzgenextreme_gen._loglogcdf>  s+    16a1f-1v7!= 	=r2   c                 L    t        j                  | j                  ||            S rH   r  r  s      r0   rl   zgenextreme_gen._pdfC  s     vvdll1a())r2   c                    t        ||k(  |dk7  z  ||fd d      }t        j                  |       }| j                  ||      }t	        j
                  |      }t	        j                  ||dk(  |t        j                   k(  z  d       t        |dk(  |t        j                   k(  z   |||fd t        j                         }t	        j                  ||dk(  |dk(  z  d       |S )Nr   c                     || z  S rH   r   rc  s     r0   r  z(genextreme_gen._logpdf.<locals>.<lambda>J  s
    !A# r2   r   r   c                     |  |z   |z
  S rH   r   )pex2lpex2lex2s      r0   r  z(genextreme_gen._logpdf.<locals>.<lambda>R  s    teemd6J r2   r  )r   rq   r  r  rJ   r   putmaskrc   )r?   rk   r  cxlogex2logpex2r  logpdfs           r0   r   zgenextreme_gen._logpdfI  s    aAF+aV5EsK2#//!Q'vvg


7Q!VbffW5s;rQw2"&&=9:!7F3J')vvg/ 	

6AFqAv.4r2   c                 N    t        j                  | j                  ||             S rH   )rJ   r   r  r  s      r0   r   zgenextreme_gen._logcdfW  s    tq!,---r2   c                 L    t        j                  | j                  ||            S rH   rJ   r   r   r  s      r0   ro   zgenextreme_gen._cdfZ  rK  r2   c                 N    t        j                  | j                  ||             S rH   rI  r  s      r0   rs   zgenextreme_gen._sf]  rJ  r2   c                     t        j                  t        j                  |              }t        ||k(  |dk7  z  ||fd |      S )Nr   c                 <    t        j                  | | z         |z  S rH   r2  rc  s     r0   r  z%genextreme_gen._ppf.<locals>.<lambda>c      !a(8'81'< r2   )rJ   r   r   r?   rw   r  rk   s       r0   rx   zgenextreme_gen._ppf`  sF    VVRVVAYJ16a1f-1v<aA 	Ar2   c                     t        j                  t        j                  |               }t	        ||k(  |dk7  z  ||fd |      S )Nr   c                 <    t        j                  | | z         |z  S rH   r2  rc  s     r0   r  z%genextreme_gen._isf.<locals>.<lambda>h  r  r2   )rJ   r   rq   r  r   r  s       r0   r{   zgenextreme_gen._isfe  sH    VVRXXqb\M""16a1f-1v<aA 	Ar2   c                    fd} |d      } |d      } |d      } |d      }t        j                  t              dk  t         j                  z  dz  dz  ||dz  z
        }t        j                  t              dk  t         j                  dz  dz  t	        j
                  t	        j                  dz  d	z         dt	        j                  d	z         z  z
        dz  z        }d
}	t        j                  t              |	k  t         t	        j
                  t	        j                  dz               z        }
t        j                  dk  t         j                  |
       }t        j                  dk  t         j                  |dz  |z        }t        dk\  ||||fd t         j                        }t        j                  t              |	dz  k  dt        j                  d      z  t        z  t         j                  dz  z  |      }t        dk\  |||||fd t         j                        }t        j                  t              |	dz  k  d|dz
        }||||fS )Nc                 :    t        j                  | z  dz         S rX   r"  )r\   r  s    r0   gz genextreme_gen._stats.<locals>.gk  s    88AEAI&&r2   r   rO   r  r  gHz>r   r:  r   +=r3  r  r  c                 X    t        j                  |       | |d|z  z   |z  z   z  |dz  z  S NrO   r  rI   )r  r>  r?  g3g2mg12s        r0   r  z'genextreme_gen._stats.<locals>.<lambda>}  s4    WWQZ"QvXr/A)AB63;N r2   r  g(\?r  r  g      пc                 <    |d|z  d||z   z  | z  z   | z  z   |dz  z  S )Nr  r  rO   r   )r>  r?  r  g4r  s        r0   r  z'genextreme_gen._stats.<locals>.<lambda>  s3     BrEArF{OB,>$>#BBFAIM r2   gq=
ףp?333333@r;  )rJ   rF  r  r   rq   r  r  r    r  r   r   r!   )r?   r  r  r>  r?  r  r  r  gam2kepsgamkr  r  sk1r  ku1r  s    `               r0   r   zgenextreme_gen._statsj  s
   	'qTqTqTqT#a&4-!BEE'C);RCZHQ$s
3"**SU3Y"7"**QW:M8M"MNqRUvUWxxAvgrxx

1q58I/J1/LMHHQXrvvu-HHQXrvvr3wu}5 eRR0O#%66	+
 XXc!fT	)2bggaj=+?q+H#N eb"b&1N#%66	+
 XXc!ft+Xs3w?!R|r2   c                     t        |t              r|j                         }t        |      }|dk  rd}nd}t        |   ||f      S )Nr   r   r  rF  r9   r%   r  r   r;   r  )r?   r@   r  r   r  s       r0   r  zgenextreme_gen._fitstart  sJ    dL)>>#D$Kq5AAw QD 11r2   c                 6   t        j                  d|dz         }d||z  z  t        j                  t        j                  ||      d|z  z  t        j
                  ||z  dz         z  d      z  }t        j                  ||z  dkD  |t         j                        S )Nr   r   r   r  r  )rJ   rD  r  rq   ry  r  rF  rc   )r?   r\   r  r
  valss        r0   r   zgenextreme_gen._munp  s    IIa11a4x"&&GGAqMR!G#bhhqsQw&77  xx!b$//r2   c                      t         d|z
  z  dz   S rX   rT  rr  s     r0   r   zgenextreme_gen._entropy  s    q1u~!!r2   )r}   r~   r   r   r]   re   r   r  rl   r   r   ro   rs   rx   r{   r   r  r   r   r  r  s   @r0   r  r    sX    %LK
=
*.*-A
A
B	20"r2   r  
genextremec                 J    d} fd} dkD  r7t        j                         dz   } dk  rDt        j                  ||d      }|S  dkD  rt        j                   d	z        d
z   }n	d  |z
  z  }t        j                  ||dd      \  }}}}|dk7  rt        d z        |d   S )af  Inverse of the digamma function (real positive arguments only).

    This function is used in the `fit` method of `gamma_gen`.
    The function uses either optimize.fsolve or optimize.newton
    to solve `sc.digamma(x) - y = 0`.  There is probably room for
    improvement, but currently it works over a wide range of y:

    >>> import numpy as np
    >>> rng = np.random.default_rng()
    >>> y = 64*rng.standard_normal(1000000)
    >>> y.min(), y.max()
    (-311.43592651416662, 351.77388222276869)
    >>> x = [_digammainv(t) for t in y]
    >>> np.abs(sc.digamma(x) - y).max()
    1.1368683772161603e-13

    gox?c                 4    t        j                  |       z
  S rH   )rq   r  r]  s    r0   rW  z_digammainv.<locals>.func  s    zz!}q  r2   g      r   r  绽|=)tolr  g-@g뭁,?r   dy=T)xtolr  r   z"_digammainv: fsolve failed, y = %rr   )rJ   r   r   newtonr  RuntimeError)r^  _emrW  x0valuer  r  rP  s   `       r0   _digammainvr    s    $ &C! 	6zVVAY_r6 OOD"%8EL	
RVVAeG_w&QBH%__T2E9=?E4d
ax?!CDD8Or2   c                        e Zd ZdZd ZddZd Zd Zd Zd Z	d Z
d	 Zd
 Zd Z fdZ eed       fd       Z xZS )	gamma_gena  A gamma continuous random variable.

    %(before_notes)s

    See Also
    --------
    erlang, expon

    Notes
    -----
    The probability density function for `gamma` is:

    .. math::

        f(x, a) = \frac{x^{a-1} e^{-x}}{\Gamma(a)}

    for :math:`x \ge 0`, :math:`a > 0`. Here :math:`\Gamma(a)` refers to the
    gamma function.

    `gamma` takes ``a`` as a shape parameter for :math:`a`.

    When :math:`a` is an integer, `gamma` reduces to the Erlang
    distribution, and when :math:`a=1` to the exponential distribution.

    Gamma distributions are sometimes parameterized with two variables,
    with a probability density function of:

    .. math::

        f(x, \alpha, \beta) =
        \frac{\beta^\alpha x^{\alpha - 1} e^{-\beta x }}{\Gamma(\alpha)}

    Note that this parameterization is equivalent to the above, with
    ``scale = 1 / beta``.

    %(after_notes)s

    %(example)s

    c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zgamma_gen._shape_info  r  r2   c                 &    |j                  ||      S rH   standard_gamma)r?   r   r   r   s       r0   r   zgamma_gen._rvs  s    **1d33r2   c                 L    t        j                  | j                  ||            S rH   r  r  s      r0   rl   zgamma_gen._pdf  rx  r2   c                 f    t        j                  |dz
  |      |z
  t        j                  |      z
  S r  )rq   rr  r  r  s      r0   r   zgamma_gen._logpdf	  s)    xx#q!A%

155r2   c                 .    t        j                  ||      S rH   r  r  s      r0   ro   zgamma_gen._cdf  r|   r2   c                 .    t        j                  ||      S rH   r  r  s      r0   rs   zgamma_gen._sf  s    ||Aq!!r2   c                 .    t        j                  ||      S rH   r  r  s      r0   rx   zgamma_gen._ppf  s    ~~a##r2   c                 .    t        j                  ||      S rH   rq   r  r  s      r0   r{   zgamma_gen._isf  s    q!$$r2   c                 @    ||dt        j                  |      z  d|z  fS )Nr   r:  r  r  s     r0   r   zgamma_gen._stats  s!    !S^SU**r2   c                 4    d }d }t        |dk  |f||      S )Nc                 j    t        j                  |       d| z
  z  | z   t        j                  |       z   S rX   rq   rT  r  r   s    r0   r  z+gamma_gen._entropy.<locals>.regular_formula  s+    66!9!$q(2::a=88r2   c                     ddt        j                  dt         j                  z        z   t        j                  |       z   z  dd| z  z  z
  | dz  dz  z
  | dz  d	z  z
  | d
z  dz  z   S )Nr   r   rO   r   r  r  r  r  r  r  r  r   r  s    r0   r  z.gamma_gen._entropy.<locals>.asymptotic_formula   sq    
 2qw/"&&);<q!a%yH#vrk"%&VRK034c63,? @r2      r  r  )r?   r   r  r  s       r0   r   zgamma_gen._entropy  s+    	9	@ !c'A5//1 	1r2   c                     t        |t              r|j                         }t        |      }dd|dz  z   z  }t        |   ||f      S )Nr  :0yE>rO   rF  r  )r?   r@   r  r   r  s       r0   r  zgamma_gen._fitstart+  sM     dL)>>#D4[Aw QD 11r2   a<          When the location is fixed by using the argument `floc`
        and `method='MLE'`, this
        function uses explicit formulas or solves a simpler numerical
        problem than the full ML optimization problem.  So in that case,
        the `optimizer`, `loc` and `scale` arguments are ignored.
        

r   c                    |j                  dd       }|j                  dd      }t        |t              s|&|j                         dk7  rt	        |   |g|i |S |j                  dd        t        |g d      }|j                  dd       }t        |       |||t        d      t        j                  |      }t        j                  |      j                         st        d      |j                         dk(  rt        j                  |      }t        j                  |      }	t        j                  ||z
  d	z        }
|||}}}||
||
d
|	z  z  }||t        j                   |	|z        }|
||	||z
  z  }|
||	|d
z  z  }|||z
  |z  }||||z  z
  }|||z
  |z  }|||fS t        j"                  ||k        rt%        d|t        j&                        |dk7  r||z
  }|j                         }|||}nt        j(                  |      t        j(                  |      j                         z
  d	z
  t        j                   d	z
  d
z  dz  z         z   dz  z  }|dz  }|dz  }t+        j,                  fd||d      }||z  }nFt        j(                  |      j                         t        j(                  |      z
  }t/        |      }|}|||fS )Nr   r,   r5   r6   r  r   r   r   r  rO   r  r  r   r  r  g333333?gffffff?c                 `    t        j                  |       t        j                  |       z
  z
  S rH   )rJ   r   rq   r  )r   r  s    r0   r  zgamma_gen.fit.<locals>.<lambda>  s!    bffQi"**Q-.G!.K r2   )disp)r7   r9   r%   r8   r;   r=   r-   r   r1   r   rJ   r   r   r   r   r  r   r  rD  rc   r   r   brentqr  )r?   r@   rA   r/   r   r,   r  r   m1m2m3r   r)   r*   r  aestxar{  r  r  r  s                      @r0   r=   zgamma_gen.fit7  s    xx%(E*t\*4!7 7;t3d3d33 	!$(=>(D)$T*>d.63E  ) * * zz${{4 $$&CDD <<>T!BB$))*BfEsAyS[U]a"f{u}QyU]b3hyS[%1*%y#X&{1u9n}cQc5= 
 66$$,wd"&&AA19 $;Dyy{ >~ FF4L266$<#4#4#66!bggqsQhAo662a4@5\5\OO$K$&4
 1HE
 t!!#bffVn4AAAE$~r2   r   )r}   r~   r   r   re   r   rl   r   ro   rs   rx   r{   r   r   r  r   r   r=   r  r  s   @r0   r  r    sc    'PE4*6!"$%+1 
2 } 5 eer2   r  r  c                   R     e Zd ZdZd Zd Z fdZ eed       fd       Z	 xZ
S )
erlang_gena  An Erlang continuous random variable.

    %(before_notes)s

    See Also
    --------
    gamma

    Notes
    -----
    The Erlang distribution is a special case of the Gamma distribution, with
    the shape parameter `a` an integer.  Note that this restriction is not
    enforced by `erlang`. It will, however, generate a warning the first time
    a non-integer value is used for the shape parameter.

    Refer to `gamma` for examples.

    c                     t        j                  t        j                  |      |k(        }|s"d|d}t        j                  |t
        d       |dkD  S )NzRThe shape parameter of the erlang distribution has been given a non-integer value .r  
stacklevelr   )rJ   r   floorwarningswarnRuntimeWarning)r?   r   allintmessages       r0   r]   zerlang_gen._argcheck  sM    q()==>EDGMM'>a@1ur2   c                 @    t        dddt        j                  fd      gS )Nr   Tr   ra   rb   rd   s    r0   re   zerlang_gen._shape_info  rf   r2   c                     t        |t              r|j                         }t        ddt	        |      dz  z   z        }t
        t        |   ||f      S )NrA  r  rO   rF  )r9   r%   r  r  r   r;   r  r  )r?   r@   r   r  s      r0   r  zerlang_gen._fitstart  sP     dL)>>#DteDk1n,-.Y/A4/@@r2   a          The Erlang distribution is generally defined to have integer values
        for the shape parameter.  This is not enforced by the `erlang` class.
        When fitting the distribution, it will generally return a non-integer
        value for the shape parameter.  By using the keyword argument
        `f0=<integer>`, the fit method can be constrained to fit the data to
        a specific integer shape parameter.r   c                 *    t        |   |g|i |S rH   )r;   r=   r?   r@   rA   r/   r  s       r0   r=   zerlang_gen.fit  s     w{4/$/$//r2   )r}   r~   r   r   r]   re   r  r   r   r=   r  r  s   @r0   r  r    s9    &CA } 5/ 0000r2   r  erlangc                   T    e Zd ZdZd Zd Zd Zd Zd ZddZ	d	 Z
d
 Zd Zd Zd Zy)gengamma_gena  A generalized gamma continuous random variable.

    %(before_notes)s

    See Also
    --------
    gamma, invgamma, weibull_min

    Notes
    -----
    The probability density function for `gengamma` is ([1]_):

    .. math::

        f(x, a, c) = \frac{|c| x^{c a-1} \exp(-x^c)}{\Gamma(a)}

    for :math:`x \ge 0`, :math:`a > 0`, and :math:`c \ne 0`.
    :math:`\Gamma` is the gamma function (`scipy.special.gamma`).

    `gengamma` takes :math:`a` and :math:`c` as shape parameters.

    %(after_notes)s

    References
    ----------
    .. [1] E.W. Stacy, "A Generalization of the Gamma Distribution",
       Annals of Mathematical Statistics, Vol 33(3), pp. 1187--1192.

    %(example)s

    c                     |dkD  |dk7  z  S r  r   )r?   r   r  s      r0   r]   zgengamma_gen._argcheck      A!q&!!r2   c                     t        dddt        j                  fd      }t        ddt        j                   t        j                  fd      }||gS rc  rb   rd  s      r0   re   zgengamma_gen._shape_info  B    UQK@UbffWbff$5~FBxr2   c                 N    t        j                  | j                  |||            S rH   r  rf  s       r0   rl   zgengamma_gen._pdf      vvdll1a+,,r2   c                 \    t        |dk7  |dkD  z  ||ffdt        j                         S )Nr   c                     t        j                  t        |            t        j                  |z  dz
  |       z   | |z  z
  t        j
                        z
  S rX   )rJ   r   r  rq   rr  r  )rk   r  r   s     r0   r  z&gengamma_gen._logpdf.<locals>.<lambda>  sG    s1v!A#'19M(M*+Q$)/13A)? r2   r  r  rf  s     ` r0   r   zgengamma_gen._logpdf  s4    16a!e,q!f@%'VVG- 	-r2   c                     ||z  }t        j                  ||      }t        j                  ||      }t        j                  |dkD  ||      S r  rq   r  r  rJ   rF  r?   rk   r   r  xcval1val2s          r0   ro   zgengamma_gen._cdf  B    T{{1b!||Ar"xxAtT**r2   Nc                 8    |j                  ||      }|d|z  z  S )Nr  r   r  )r?   r   r  r   r   r  s         r0   r   zgengamma_gen._rvs  s%    '''52a4yr2   c                     ||z  }t        j                  ||      }t        j                  ||      }t        j                  |dkD  ||      S r  r  r  s          r0   rs   zgengamma_gen._sf  r  r2   c                     t        j                  ||      }t        j                  ||      }t        j                  |dkD  ||      d|z  z  S r  rq   r  r  rJ   rF  r?   rw   r   r  r  r  s         r0   rx   zgengamma_gen._ppf$  B    ~~a#q!$xxAtT*SU33r2   c                     t        j                  ||      }t        j                  ||      }t        j                  |dkD  ||      d|z  z  S r  r  r  s         r0   r{   zgengamma_gen._isf)  r  r2   c                 :    t        j                  ||dz  |z        S r  )rq   r  )r?   r\   r   r  s       r0   r   zgengamma_gen._munp.  s    wwq!C%'""r2   c                 :    d }d }t        |dk\  ||f||      }|S )Nc                     t        j                  |       }| d|z
  z  ||z  z   }t        j                  |       t        j                  t        |            z
  }||z   }|S rX   )rq   rT  r  rJ   r   r  )r   r  r7  ABr  s         r0   r  z&gengamma_gen._entropy.<locals>.regular3  sQ    &&)CQWa'A

1s1v.AAAHr2   c                 :   t         j                         t        j                  |       dz  z
  t        j                  t        j                  |            z
  | dz  dz  z   | dz  dz  z
  t        j                  |       | dz  dz  z
  | dz  dz  z
  | dz  d	z  z   |z  z   S )
NrO   r3  r  r  r  r  r  r  r  )r   r   rJ   r   r  )r   r  s     r0   
asymptoticz)gengamma_gen._entropy.<locals>.asymptotic:  s    MMObffQik1ffRVVAY'(+,c61*5893{CvvayAsFA:-C;q#vslJAMN Or2         i@r|  r  r  )r?   r   r  r  r$  r  s         r0   r   zgengamma_gen._entropy2  s,    		O qCx!Q:'Br2   r   )r}   r~   r   r   r]   re   rl   r   ro   r   rs   rx   r{   r   r   r   r2   r0   r  r    s>    >"
--++4
4
#r2   r  gengammac                   4    e Zd ZdZd Zd Zd Zd Zd Zd Z	y)	genhalflogistic_gena  A generalized half-logistic continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `genhalflogistic` is:

    .. math::

        f(x, c) = \frac{2 (1 - c x)^{1/(c-1)}}{[1 + (1 - c x)^{1/c}]^2}

    for :math:`0 \le x \le 1/c`, and :math:`c > 0`.

    `genhalflogistic` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zgenhalflogistic_gen._shape_info]  r  r2   c                 $    | j                   d|z  fS r  r  rr  s     r0   r   z genhalflogistic_gen._get_support`  s    vvs1u}r2   c                 x    d|z  }t        j                  d||z  z
        }||dz
  z  }||z  }d|z  d|z   dz  z  S r  rJ   r   )r?   rk   r  limitr  tmp0tmp2s          r0   rl   zgenhalflogistic_gen._pdfc  sP     Ajj1Q3U1W~Cxv4!##r2   c                 b    d|z  }t        j                  d||z  z
        }||z  }d|z
  d|z   z  S r   r-  )r?   rk   r  r.  r  r0  s         r0   ro   zgenhalflogistic_gen._cdfl  s=    Ajj1Q3U|DQtV$$r2   c                 0    d|z  dd|z
  d|z   z  |z  z
  z  S r   r   r  s      r0   rx   zgenhalflogistic_gen._ppfr  s'    1ua#a%#a%1,,--r2   c                 D    dd|z  dz   t        j                  d      z  z
  S r  r+  rr  s     r0   r   zgenhalflogistic_gen._entropyu  s"    AaCE266!9$$$r2   N)
r}   r~   r   r   re   r   rl   ro   rx   r   r   r2   r0   r)  r)  G  s&    *E$%.%r2   r)  genhalflogisticc                   p     e Zd ZdZd Zd Z fdZd Zd Zd e	d               Z
d	 Zd
 ZddZd Z xZS )genhyperbolic_genu  A generalized hyperbolic continuous random variable.

    %(before_notes)s

    See Also
    --------
    t, norminvgauss, geninvgauss, laplace, cauchy

    Notes
    -----
    The probability density function for `genhyperbolic` is:

    .. math::

        f(x, p, a, b) =
            \frac{(a^2 - b^2)^{p/2}}
            {\sqrt{2\pi}a^{p-1/2}
            K_p\Big(\sqrt{a^2 - b^2}\Big)}
            e^{bx} \times \frac{K_{p - 1/2}
            (a \sqrt{1 + x^2})}
            {(\sqrt{1 + x^2})^{1/2 - p}}

    for :math:`x, p \in ( - \infty; \infty)`,
    :math:`|b| < a` if :math:`p \ge 0`,
    :math:`|b| \le a` if :math:`p < 0`.
    :math:`K_{p}(.)` denotes the modified Bessel function of the second
    kind and order :math:`p` (`scipy.special.kv`)

    `genhyperbolic` takes ``p`` as a tail parameter,
    ``a`` as a shape parameter,
    ``b`` as a skewness parameter.

    %(after_notes)s

    The original parameterization of the Generalized Hyperbolic Distribution
    is found in [1]_ as follows

    .. math::

        f(x, \lambda, \alpha, \beta, \delta, \mu) =
           \frac{(\gamma/\delta)^\lambda}{\sqrt{2\pi}K_\lambda(\delta \gamma)}
           e^{\beta (x - \mu)} \times \frac{K_{\lambda - 1/2}
           (\alpha \sqrt{\delta^2 + (x - \mu)^2})}
           {(\sqrt{\delta^2 + (x - \mu)^2} / \alpha)^{1/2 - \lambda}}

    for :math:`x \in ( - \infty; \infty)`,
    :math:`\gamma := \sqrt{\alpha^2 - \beta^2}`,
    :math:`\lambda, \mu \in ( - \infty; \infty)`,
    :math:`\delta \ge 0, |\beta| < \alpha` if :math:`\lambda \ge 0`,
    :math:`\delta > 0, |\beta| \le \alpha` if :math:`\lambda < 0`.

    The location-scale-based parameterization implemented in
    SciPy is based on [2]_, where :math:`a = \alpha\delta`,
    :math:`b = \beta\delta`, :math:`p = \lambda`,
    :math:`scale=\delta` and :math:`loc=\mu`

    Moments are implemented based on [3]_ and [4]_.

    For the distributions that are a special case such as Student's t,
    it is not recommended to rely on the implementation of genhyperbolic.
    To avoid potential numerical problems and for performance reasons,
    the methods of the specific distributions should be used.

    References
    ----------
    .. [1] O. Barndorff-Nielsen, "Hyperbolic Distributions and Distributions
       on Hyperbolae", Scandinavian Journal of Statistics, Vol. 5(3),
       pp. 151-157, 1978. https://www.jstor.org/stable/4615705

    .. [2] Eberlein E., Prause K. (2002) The Generalized Hyperbolic Model:
        Financial Derivatives and Risk Measures. In: Geman H., Madan D.,
        Pliska S.R., Vorst T. (eds) Mathematical Finance - Bachelier
        Congress 2000. Springer Finance. Springer, Berlin, Heidelberg.
        :doi:`10.1007/978-3-662-12429-1_12`

    .. [3] Scott, David J, Würtz, Diethelm, Dong, Christine and Tran,
       Thanh Tam, (2009), Moments of the generalized hyperbolic
       distribution, MPRA Paper, University Library of Munich, Germany,
       https://EconPapers.repec.org/RePEc:pra:mprapa:19081.

    .. [4] E. Eberlein and E. A. von Hammerstein. Generalized hyperbolic
       and inverse Gaussian distributions: Limiting cases and approximation
       of processes. FDM Preprint 80, April 2003. University of Freiburg.
       https://freidok.uni-freiburg.de/fedora/objects/freidok:7974/datastreams/FILE1/content

    %(example)s

    c                     t        j                  t        j                  |      |k  |dk\        t        j                  t        j                  |      |k  |dk        z  S r  )rJ   logical_andr  )r?   r  r   r   s       r0   r]   zgenhyperbolic_gen._argcheck  sH    rvvay1}a1f5..aQ78 	9r2   c                     t        ddt        j                   t        j                  fd      }t        dddt        j                  fd      }t        ddt        j                   t        j                  fd      }|||gS )Nr  Fr  r   r   ra   r   rb   )r?   iprb  rc  s       r0   re   zgenhyperbolic_gen._shape_info  sd    UbffWbff$5~FUQK?UbffWbff$5~FB|r2   c                 &    t         |   |d      S )N)r   r   r   rF  r  r  s     r0   r  zgenhyperbolic_gen._fitstart  s     w K 88r2   c                 D    t         j                  d        } |||||      S )Nc                 2    t        j                  | |||      S rH   )r   genhyperbolic_logpdfrk   r  r   r   s       r0   _logpdf_singlez1genhyperbolic_gen._logpdf.<locals>._logpdf_single  s    ..q!Q::r2   rJ   	vectorize)r?   rk   r  r   r   r@  s         r0   r   zgenhyperbolic_gen._logpdf  s-     
	; 
	; aAq))r2   c                 D    t         j                  d        } |||||      S )Nc                 2    t        j                  | |||      S rH   )r   genhyperbolic_pdfr?  s       r0   _pdf_singlez+genhyperbolic_gen._pdf.<locals>._pdf_single  s    ++Aq!Q77r2   rA  )r?   rk   r  r   r   rF  s         r0   rl   zgenhyperbolic_gen._pdf  s-     
	8 
	8 1aA&&r2   c                 t    t        j                  | j                  t              t         j                  g      S )Notypes)rJ   rB  __get__objectfloat64)rW  s    r0   r  zgenhyperbolic_gen.<lambda>  s     ",,t||F3RZZLI r2   c                    t        j                  |||gt              j                  j	                  t        j
                        }t        j                  t        d|      }t        j                  ||z   ||z
  z        }||z  t        j                  |dz   |      z  t        j                  ||      z  }d}	d}
| |cxk  r|k  r?n n<t        j                  || ||	|
      d   t        j                  ||||	|
      d   z   }nt        j                  || ||	|
      d   }t        j                  |      rd}t        j                   |t"        d       t%        d	t'        d
|            S )z
        Integrate the pdf of the genhyberbolic distribution from x0 to x1.
        This is a private function used by _cdf() and _sf() only; either x0
        will be -inf or x1 will be inf.
        _genhyperbolic_pdfr   r  r   )epsrelepsabszdInfinite values encountered in scipy.special.kve. Values replaced by NaN to avoid incorrect results.r  r  r   r   )rJ   arrayrA  ctypesdata_asc_void_pr   from_cythonr   r   rq   kvr   quadisnanr  r  r  maxr@  )r  x1r  r   r   	user_datallcr  r   rO  rP  intgrlrS   s                r0   _integrate_pdfz genhyperbolic_gen._integrate_pdf  sJ    HHaAY.55==fooN	**63G+46GGQUQUO$sRUU1q5!_$ruuQ{2>r>  nnS"d,26CCDF!sD".4VEEFHHF
 ^^CR+1&BBCEF88FHCMM#~!<3C())r2   c                 J    | j                  t        j                   ||||      S rH   r^  rJ   rc   r?   rk   r  r   r   s        r0   ro   zgenhyperbolic_gen._cdf  s!    ""BFF7Aq!Q77r2   c                 H    | j                  |t        j                  |||      S rH   r`  ra  s        r0   rs   zgenhyperbolic_gen._sf"  s    ""1bffaA66r2   c                 R   t        j                  |d      t        j                  |d      z
  }t        j                  |d      }t        j                  |d      }t        j                  |||||      }	t        j                  ||      }
||	z  t        j
                  |	      |
z  z   S )NrO   r   r  )r  r   r*   r   r   r  )rJ   float_powergeninvgaussr  r   r   )r?   r  r   r   r   r   r  r  r  gignormsts              r0   r   zgenhyperbolic_gen._rvs%  s    
 ^^Aq!BNN1a$88^^B$^^B&oo%   t,?3w...r2   c                    t        j                  |||      \  }}}t        j                  |d      t        j                  |d      z
  }t        j                  |d      }t        j                  dd      t        j                  |d      z  }t        j                  ddd      }|j	                  |j
                  d|j                  z  z         }t        j                  ||z   |      \  }}}	}
fd	|||	|
fD        \  }}}}||z  |z  }||z  t        j                  |d      t        j                  |d      z  |t        j                  |d      z
  z  z   }t        j                  |d
      t        j                  |d
      z  |d
|z  |z  t        j                  d      z  z
  dt        j                  |d
      z  z   z  d
|z  t        j                  |d      z  |t        j                  |d      z
  z  z   }|t        j                  |d      z  }t        j                  |d      t        j                  |d      z  |d|	z  |z  t        j                  d      z  z
  d|z  t        j                  |d      z  t        j                  d      z  z   d
t        j                  |d      z  z
  z  t        j                  |d      t        j                  |d
      z  d|z  d|z  |z  t        j                  d      z  z
  dt        j                  |d
      z  z   z  z   d
t        j                  |d      z  |z  z   }|t        j                  |d      z  d
z
  }||||fS )NrO   r   r   r  r   r  r:  )r   c              3   (   K   | ]	  }|z    y wrH   r   ).0r   b0s     r0   	<genexpr>z+genhyperbolic_gen._stats.<locals>.<genexpr>F  s     ;Q!b&;s   r  r	  r:  r  r  r  )	rJ   r  rd  linspacerE  shaper"  rq   rV  )r?   r  r   r   r  r  integersb1b2b3b4r1r2r3r4r  r  m3er  m4er
  rk  s                        @r0   r   zgenhyperbolic_gen._stats:  s    %%aA.1a^^Aq!BNN1a$88^^B$^^Aq!BNN2s$;;;;q!Q'##HNNTAFF]$BCUU1x<4BB;2r2r*:;BBFRKGbnnQ*R^^B-BB"..Q'') ) 	

 NN1a 2>>"a#88!b&2+r2 666A&&'( EBNN2q))"..Q'')) 	 "..G,,NN1a 2>>"a#88!b&2+r3 777VbnnR++bnnR.EEFA&&'( NN1a 2>>"a#88Vb2glR^^B%<<<A&&'(	( r1%%*+ 	 "..B''!+!Qzr2   r   )r}   r~   r   r   r]   re   r  r   rl   staticmethodr^  ro   rs   r   r   r  r  s   @r0   r6  r6  |  sV    Wr99
*' J*  J*@87/*'r2   r6  genhyperbolicc                   @    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zy
)gompertz_genaq  A Gompertz (or truncated Gumbel) continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `gompertz` is:

    .. math::

        f(x, c) = c \exp(x) \exp(-c (e^x-1))

    for :math:`x \ge 0`, :math:`c > 0`.

    `gompertz` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zgompertz_gen._shape_info}  r  r2   c                 L    t        j                  | j                  ||            S rH   r  r  s      r0   rl   zgompertz_gen._pdf  rx  r2   c                 d    t        j                  |      |z   |t        j                  |      z  z
  S rH   r  r  s      r0   r   zgompertz_gen._logpdf  s%    vvay1}q288A;..r2   c                 \    t        j                  | t        j                  |      z         S rH   r2  r  s      r0   ro   zgompertz_gen._cdf  s#    !bhhqk)***r2   c                 `    t        j                  d|z  t        j                  |       z        S r6  r   r  s      r0   rx   zgompertz_gen._ppf  s$    xxq288QB</00r2   c                 Z    t        j                  | t        j                  |      z        S rH   r  r  s      r0   rs   zgompertz_gen._sf  s     vvqb288A;&''r2   c                 Z    t        j                  t        j                  |       |z        S rH   r  r?   r  r  s      r0   r{   zgompertz_gen._isf  s    xx
1%%r2   c                 x    dt        j                  |      z
  t        j                  j	                  |      |z  z
  S r  )rJ   r   rq   _ufuncs_scaled_exp1rr  s     r0   r   zgompertz_gen._entropy  s-    RVVAY!8!8!;A!===r2   Nr}   r~   r   r   re   rl   r   ro   rx   rs   r{   r   r   r2   r0   r}  r}  g  s0    *E*/+1(&>r2   r}  gompertzc                     t        j                  |       } t        j                  |      }|j                         }t        j                  ||z
        }t        j                  | |      S )N)weights)rJ   r   rY  r   average)rk   
logweightsmaxlogwr  s       r0   _average_with_log_weightsr    sM    


1AJ'JnnGffZ')*G::a))r2   c                   r    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zd Ze ee      d               Zy)gumbel_r_gena  A right-skewed Gumbel continuous random variable.

    %(before_notes)s

    See Also
    --------
    gumbel_l, gompertz, genextreme

    Notes
    -----
    The probability density function for `gumbel_r` is:

    .. math::

        f(x) = \exp(-(x + e^{-x}))

    The Gumbel distribution is sometimes referred to as a type I Fisher-Tippett
    distribution.  It is also related to the extreme value distribution,
    log-Weibull and Gompertz distributions.

    %(after_notes)s

    %(example)s

    c                     g S rH   r   rd   s    r0   re   zgumbel_r_gen._shape_info  r   r2   c                 J    t        j                  | j                  |            S rH   r  r   s     r0   rl   zgumbel_r_gen._pdf      vvdll1o&&r2   c                 6    | t        j                  |       z
  S rH   r.  r   s     r0   r   zgumbel_r_gen._logpdf  s    rBFFA2Jr2   c                 V    t        j                  t        j                  |              S rH   r.  r   s     r0   ro   zgumbel_r_gen._cdf  s    vvrvvqbzk""r2   c                 0    t        j                  |        S rH   r.  r   s     r0   r   zgumbel_r_gen._logcdf  s    r
{r2   c                 V    t        j                  t        j                  |              S rH   r+  r   s     r0   rx   zgumbel_r_gen._ppf  s    q	z"""r2   c                 X    t        j                  t        j                  |               S rH   r  r   s     r0   rs   zgumbel_r_gen._sf  s     "&&!*%%%r2   c                 X    t        j                  t        j                  |               S rH   rJ   r   r  r  s     r0   r{   zgumbel_r_gen._isf  s     !}%%%r2   c                     t         t        j                  t        j                  z  dz  dt        j                  d      z  t        j                  dz  z  t        z  dfS )Nr:  r  r  r  r  r    rJ   r   r   r!   rd   s    r0   r   zgumbel_r_gen._stats  s?    ruuRUU{32771:beeQh(>(GOOr2   c                     t         dz   S r  rT  rd   s    r0   r   zgumbel_r_gen._entropy  s    {r2   c                    t        | ||      \  }}fd}||} ||      |fS |	|fdnfd|j                  dd      }|dz  |dz  }
}	fd} ||	|
      sE|	dkD  s|
t        j                  k  r-|	dz  }	|
dz  }
 ||	|
      s|	dkD  r|
t        j                  k  r-t	        j
                  |	|
fd	d	
      }|j                  }||n ||      |fS )Nc                 |    |  t        j                   | z        t        j                  t	                    z
  z  S rH   )rq   	logsumexprJ   r   r  )r*   r@   s    r0   get_loc_from_scalez,gumbel_r_gen.fit.<locals>.get_loc_from_scale  s1    6R\\4%%-8266#d);LLMMr2   c                     z
  t        j                  z
  | z        z  z   }t              | z   z  }|j                         |z
  S rH   )rJ   r   r  r  )r*   term1term2r@   r)   s      r0   rW  zgumbel_r_gen.fit.<locals>.func  sK     4Z2663:2F+GG$NEIu5E 99;..r2   c                 V     | z  }t        |      }j                         |z
  | z
  S )N)r  )r  r   )r*   sdatawavgr@   s      r0   rW  zgumbel_r_gen.fit.<locals>.func  s0    !EEME4TeLD99;-55r2   r*   r   rO   c                 r    t        j                   |             t        j                   |            k7  S rH   rI   )rL   rM   rW  s     r0   rN   z0gumbel_r_gen.fit.<locals>.interval_contains_root  s-    V-V-. /r2   r   r  )r  rtolr  )r  r7   rJ   rc   r   r&   r  )r?   r@   rA   r/   r   r   r  r*   brack_startrL   rM   rN   resrW  r)   s    `           @@r0   r=   zgumbel_r_gen.fit  s    9t9=tEdF	N  E$U+C\ EzU /6 ((7A.K(1_kAoFF
/ .ff=
frvvo!! .ff=
frvvo &&tff5E,1?CHHE*$0B50ICEzr2   N)r}   r~   r   r   re   rl   r   ro   r   rx   rs   r{   r   r   rE   r
   r   r=   r   r2   r0   r  r    s]    2'##&&P M*@ + @r2   r  gumbel_rc                   r    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zd Ze ee      d               Zy)gumbel_l_gena  A left-skewed Gumbel continuous random variable.

    %(before_notes)s

    See Also
    --------
    gumbel_r, gompertz, genextreme

    Notes
    -----
    The probability density function for `gumbel_l` is:

    .. math::

        f(x) = \exp(x - e^x)

    The Gumbel distribution is sometimes referred to as a type I Fisher-Tippett
    distribution.  It is also related to the extreme value distribution,
    log-Weibull and Gompertz distributions.

    %(after_notes)s

    %(example)s

    c                     g S rH   r   rd   s    r0   re   zgumbel_l_gen._shape_info?  r   r2   c                 J    t        j                  | j                  |            S rH   r  r   s     r0   rl   zgumbel_l_gen._pdfB  r  r2   c                 2    |t        j                  |      z
  S rH   r.  r   s     r0   r   zgumbel_l_gen._logpdfF  rt  r2   c                 V    t        j                  t        j                  |              S rH   r  r   s     r0   ro   zgumbel_l_gen._cdfI  s    "&&)$$$r2   c                 V    t        j                  t        j                  |              S rH   rJ   r   rq   r  r   s     r0   rx   zgumbel_l_gen._ppfL  s    vvrxx|m$$r2   c                 .    t        j                  |       S rH   r.  r   s     r0   r   zgumbel_l_gen._logsfO  r8  r2   c                 T    t        j                  t        j                  |             S rH   r.  r   s     r0   rs   zgumbel_l_gen._sfR      vvrvvayj!!r2   c                 T    t        j                  t        j                  |             S rH   r+  r   s     r0   r{   zgumbel_l_gen._isfU  r  r2   c                     t          t        j                  t        j                  z  dz  dt        j                  d      z  t        j                  dz  z  t        z  dfS )Nr:  r  r  r  r  rd   s    r0   r   zgumbel_l_gen._statsX  sF    wbeeC2771:~beeQh&/8 	8r2   c                     t         dz   S r  rT  rd   s    r0   r   zgumbel_l_gen._entropy\  s    {r2   c                     |j                  d      	|d    |d<   t        j                  t        j                  |       g|i |\  }}| |fS )Nr   )r7   r  r=   rJ   r   )r?   r@   rA   r/   loc_rscale_rs         r0   r=   zgumbel_l_gen.fit_  sV     88F' L=DL",,

4(8'8H4H4Hwvwr2   N)r}   r~   r   r   re   rl   r   ro   rx   r   rs   r{   r   r   rE   r
   r   r=   r   r2   r0   r  r  $  sZ    4'%%""8 M* + r2   r  gumbel_lc                   x     e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Ze ee       fd              Z xZS )halfcauchy_gena  A Half-Cauchy continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `halfcauchy` is:

    .. math::

        f(x) = \frac{2}{\pi (1 + x^2)}

    for :math:`x \ge 0`.

    %(after_notes)s

    %(example)s

    c                     g S rH   r   rd   s    r0   re   zhalfcauchy_gen._shape_info  r   r2   c                 :    dt         j                  z  d||z  z   z  S r  r(  r   s     r0   rl   zhalfcauchy_gen._pdf  rz  r2   c                     t        j                  dt         j                  z        t        j                  ||z        z
  S r7  rJ   r   r   rq   r  r   s     r0   r   zhalfcauchy_gen._logpdf  s*    vvc"%%i 288AaC=00r2   c                 T    dt         j                  z  t        j                  |      z  S r7  r|  r   s     r0   ro   zhalfcauchy_gen._cdf  s    255y1%%r2   c                 T    t        j                  t         j                  dz  |z        S r  r  r   s     r0   rx   zhalfcauchy_gen._ppf  s    vvbeeAgai  r2   c                 V    dt         j                  z  t        j                  d|      z  S Nr   r   )rJ   r   r  r   s     r0   rs   zhalfcauchy_gen._sf  s     255y2::a+++r2   c                 Z    dt        j                  t         j                  |z  dz        z  S r  r  r  s     r0   r{   zhalfcauchy_gen._isf  s"    266"%%'!)$$$r2   c                 ~    t         j                  t         j                  t         j                  t         j                  fS rH   r  rd   s    r0   r   zhalfcauchy_gen._stats  r  r2   c                 N    t        j                  dt         j                  z        S r  r   rd   s    r0   r   zhalfcauchy_gen._entropy  r  r2   c                    |j                  dd      rt        
|   |g|i |S t        | |||      \  }}}t	        j
                  |      }|$||k  rt        d|t        j                        |}n|}d }||}	||	fS  |||      }	||	fS )Nr  F
halfcauchyr  c                     || z
  }|j                   t        j                  |      fd}t        j                  d      j                  dz  }t        ||t        j                  |      f      }|j                  S )Nc                 P    | dz  z   }dt        j                  |z        z  z
  S r  rJ   r  )r*   denominatorr\   shifted_data_squareds     r0   fun_to_solvez<halfcauchy_gen.fit.<locals>.find_scale.<locals>.fun_to_solve  s1    #Qh)==266"6{"BCCaGGr2   r   r   r  )r   rJ   squarefinfotinyr&   rY  r  )r)   r@   shifted_datar  smallr  r\   r  s         @@r0   
find_scalez&halfcauchy_gen.fit.<locals>.find_scale  sg    #:L		A#%99\#: H HHSM&&+ElUBFF<<P4QRC88Or2   r-   r;   r=   r  rJ   r@  rD  rc   )r?   r@   rA   r/   r   r   rB  r)   r  r*   r  s             r0   r=   zhalfcauchy_gen.fit  s     88J&7;t3d3d338t9=tEdF 66$<$"<t266JJC C	 E Ez sD)EEzr2   )r}   r~   r   r   re   rl   r   ro   rx   rs   r{   r   r   rE   r
   r   r=   r  r  s   @r0   r  r  s  sV    &#1&!,%. M*% + %r2   r  r  c                   x     e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Ze ee       fd              Z xZS )halflogistic_gena  A half-logistic continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `halflogistic` is:

    .. math::

        f(x) = \frac{ 2 e^{-x} }{ (1+e^{-x})^2 }
             = \frac{1}{2} \text{sech}(x/2)^2

    for :math:`x \ge 0`.

    %(after_notes)s

    References
    ----------
    .. [1] Asgharzadeh et al (2011). "Comparisons of Methods of Estimation for the
           Half-Logistic Distribution". Selcuk J. Appl. Math. 93-108.

    %(example)s

    c                     g S rH   r   rd   s    r0   re   zhalflogistic_gen._shape_info  r   r2   c                 J    t        j                  | j                  |            S rH   r  r   s     r0   rl   zhalflogistic_gen._pdf  s     vvdll1o&&r2   c                     t        j                  d      |z
  dt        j                  t        j                  |             z  z
  S r   )rJ   r   rq   r  r   r   s     r0   r   zhalflogistic_gen._logpdf  s1    vvay1}rBHHRVVQBZ$8888r2   c                 2    t        j                  |dz        S r7  )rJ   tanhr   s     r0   ro   zhalflogistic_gen._cdf  s    wwqu~r2   c                 2    dt        j                  |      z  S r  rJ   arctanhr   s     r0   rx   zhalflogistic_gen._ppf  s    Ar2   c                 4    dt        j                  |       z  S r  rq   expitr   s     r0   rs   zhalflogistic_gen._sf  s    288QB<r2   c                 ,    t        |dk  |fd d       S )Nr   c                 4    t        j                  d| z         S r  rq   logitr   s    r0   r  z'halflogistic_gen._isf.<locals>.<lambda>   s    RXXcAg%6$6 r2   c                 8    dt        j                  d| z
        z  S r  r  r   s    r0   r  z'halflogistic_gen._isf.<locals>.<lambda>  s    qAE):': r2   r  r  r   s     r0   r{   zhalflogistic_gen._isf  s    !c'A56:< 	<r2   c                 p   |dk(  rdt        j                  d      z  S |dk(  r$t         j                  t         j                  z  dz  S |dk(  r	dt        z  S |dk(  rdt         j                  dz  z  dz  S ddt	        d	d|z
        z
  z  t        j                  |dz         z  t        j                  |d      z  S )
Nr   rO   r;  r  r	  r  r  rN  r   )rJ   r   r   r!   r  rq   r  rO  r[   s     r0   r   zhalflogistic_gen._munp  s    6RVVAY;655;s?"6V8O6RUUAX:$$!CQqSM/"288AaC=0A>>r2   c                 2    dt        j                  d      z
  S r  r+  rd   s    r0   r   zhalflogistic_gen._entropy  r,  r2   c                    |j                  dd      rt        
|   |g|i |S t        | |||      \  }}}d }t	        j
                  |      }|$||k  rt        d|t        j                        |}n|}||n |||      }	||	fS )Nr  Fc                    | j                   d   }t        j                  | d      }t        j                  d|dz         |dz   z  }d|z
  }d|z   }|d|z  |z  t        j                  ||z        z  z
  }d|z  |z  }||z
  }dt        j
                  |dd  |dd  z        z  }	dt        j
                  |dd  |dd  dz  z        z  }
|	t        j                  |	dz  d|z  |
z  z         z   d|z  z  }d}d}|j                         }||kD  rN|t        j                  | |z        z  }|d|z  |j                         z  z
  }t        ||z
  |z        }|}||kD  rN|S )	Nr   r  r   r   rO   r'  r  r  )rn  rJ   sortrD  r   r  r   r   rq   r  r  )r@   r)   n_observationssorted_datar  rw   pp1r  rg  r"  Cr*   r  relative_residualshifted_meansum_term	scale_news                    r0   r  z(halflogistic_gen.fit.<locals>.find_scale  s    "ZZ]N''$Q/K		!^a/0.12DEAAAa%Ca#sQw77E7S=D%+KBFF59{12677ABFF48k!"oq&8899A"''!Q$^);a)?"?@@.(*E D !&++-L $d*&;,u2D)EE(1^+;hlln+LL	$'):E(A$B!!	 $d*
 Lr2   halflogisticr  r  )r?   r@   rA   r/   r   r   r  rB  r)   r*   r  s             r0   r=   zhalflogistic_gen.fit  s     88J&7;t3d3d338t9=tEdF	D 66$<$">RVVLLC C !,*T32GEzr2   )r}   r~   r   r   re   rl   r   ro   rx   rs   r{   r   r   rE   r
   r   r=   r  r  s   @r0   r  r    sV    2'
9 <
	? M*6 + 6r2   r  r  c                        e Zd ZdZd ZddZd Zd Zd Zd Z	d Z
d	 Zd
 Zd Ze ee       fd              Z xZS )halfnorm_genaF  A half-normal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `halfnorm` is:

    .. math::

        f(x) = \sqrt{2/\pi} \exp(-x^2 / 2)

    for :math:`x >= 0`.

    `halfnorm` is a special case of `chi` with ``df=1``.

    %(after_notes)s

    %(example)s

    c                     g S rH   r   rd   s    r0   re   zhalfnorm_gen._shape_infoe  r   r2   c                 8    t        |j                  |            S Nr  r  r   s      r0   r   zhalfnorm_gen._rvsh  s    <//T/:;;r2   c                     t        j                  dt         j                  z        t        j                  | |z  dz        z  S r7  rJ   r   r   r   r   s     r0   rl   zhalfnorm_gen._pdfk  s1    wws255y!"&&!Ac"222r2   c                 f    dt        j                  dt         j                  z        z  ||z  dz  z
  S Nr   r   r   r   s     r0   r   zhalfnorm_gen._logpdfo  s+    RVVCI&&1S00r2   c                 X    t        j                  |t        j                  d      z        S r  rq   r  rJ   r   r   s     r0   ro   zhalfnorm_gen._cdfr  s    vva"''!*n%%r2   c                 $    t        d|z   dz        S r  r   r   s     r0   rx   zhalfnorm_gen._ppfu  s    !A#s##r2   c                     dt        |      z  S r  r   r   s     r0   rs   zhalfnorm_gen._sfx  s    8A;r2   c                     t        |dz        S r  r   r  s     r0   r{   zhalfnorm_gen._isf{  s    1~r2   c                 P   t        j                  dt         j                  z        ddt         j                  z  z
  t        j                  d      dt         j                  z
  z  t         j                  dz
  dz  z  dt         j                  dz
  z  t         j                  dz
  dz  z  fS )Nr   r   rO   r  r  r'  r  rJ   r   r   rd   s    r0   r   zhalfnorm_gen._stats~  sx    BEE	"#bee)
AbeeG$beeAg^32557RUU1WqL(* 	*r2   c                 Z    dt        j                  t         j                  dz        z  dz   S r  r   rd   s    r0   r   zhalfnorm_gen._entropy  s#    266"%%)$$S((r2   c                 :   |j                  dd      rt        	|   |g|i |S t        | |||      \  }}}t	        j
                  |      }|$||k  rt        d|t        j                        |}n|}||}||fS t        j                  |d|      dz  }||fS )Nr  Fhalfnormr  rO   )ordercenterr   )
r-   r;   r=   r  rJ   r@  rD  rc   r  moment)
r?   r@   rA   r/   r   r   rB  r)   r*   r  s
            r0   r=   zhalfnorm_gen.fit  s     88J&7;t3d3d338t9=tEdF 66$<$":THHCCE Ez LLQs;S@EEzr2   r   )r}   r~   r   r   re   r   rl   r   ro   rx   rs   r{   r   r   rE   r
   r   r=   r  r  s   @r0   r   r   O  s[    *<31&$*) M* + r2   r   r  c                   @    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zy
)hypsecant_gena  A hyperbolic secant continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `hypsecant` is:

    .. math::

        f(x) = \frac{1}{\pi} \text{sech}(x)

    for a real number :math:`x`.

    %(after_notes)s

    %(example)s

    c                     g S rH   r   rd   s    r0   re   zhypsecant_gen._shape_info  r   r2   c                 T    dt         j                  t        j                  |      z  z  S r  )rJ   r   coshr   s     r0   rl   zhypsecant_gen._pdf  s    BEE"''!*$%%r2   c                 z    dt         j                  z  t        j                  t        j                  |            z  S r7  rJ   r   r}  r   r   s     r0   ro   zhypsecant_gen._cdf  s&    255y266!9---r2   c                 z    t        j                  t        j                  t         j                  |z  dz              S r7  rJ   r   r  r   r   s     r0   rx   zhypsecant_gen._ppf  s&    vvbffRUU1WS[)**r2   c                 |    dt         j                  z  t        j                  t        j                  |             z  S r7  r  r   s     r0   rs   zhypsecant_gen._sf  s(    255y2661":...r2   c                 |    t        j                  t        j                  t         j                  |z  dz               S r7  r  r   s     r0   r{   zhypsecant_gen._isf  s)    rvvbeeAgck*+++r2   c                 R    dt         j                  t         j                  z  dz  ddfS )Nr   r  rO   r(  rd   s    r0   r   zhypsecant_gen._stats  s!    "%%+a-A%%r2   c                 N    t        j                  dt         j                  z        S r  r   rd   s    r0   r   zhypsecant_gen._entropy  r  r2   N)r}   r~   r   r   re   rl   ro   rx   rs   r{   r   r   r   r2   r0   r  r    s/    &&.+/,&r2   r  	hypsecantc                   (    e Zd ZdZd Zd Zd Zd Zy)gausshyper_gena_  A Gauss hypergeometric continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `gausshyper` is:

    .. math::

        f(x, a, b, c, z) = C x^{a-1} (1-x)^{b-1} (1+zx)^{-c}

    for :math:`0 \le x \le 1`, :math:`a,b > 0`, :math:`c` a real number,
    :math:`z > -1`, and :math:`C = \frac{1}{B(a, b) F[2, 1](c, a; a+b; -z)}`.
    :math:`F[2, 1]` is the Gauss hypergeometric function
    `scipy.special.hyp2f1`.

    `gausshyper` takes :math:`a`, :math:`b`, :math:`c` and :math:`z` as shape
    parameters.

    %(after_notes)s

    References
    ----------
    .. [1] Armero, C., and M. J. Bayarri. "Prior Assessments for Prediction in
           Queues." *Journal of the Royal Statistical Society*. Series D (The
           Statistician) 43, no. 1 (1994): 139-53. doi:10.2307/2348939

    %(example)s

    c                 0    |dkD  |dkD  z  ||k(  z  |dkD  z  S )Nr   r  r   )r?   r   r   r  r  s        r0   r]   zgausshyper_gen._argcheck  s'    A!a% AF+q2v66r2   c                    t        dddt        j                  fd      }t        dddt        j                  fd      }t        ddt        j                   t        j                  fd      }t        dddt        j                  fd      }||||gS )	Nr   Fr   r  r   r  r  r  rb   )r?   rb  rc  r  izs        r0   re   zgausshyper_gen._shape_info  sx    UQK@UQK@UbffWbff$5~FURL.ABBr2   c                     t        j                  ||      t        j                  ||||z   |       z  }d|z  ||dz
  z  z  d|z
  |dz
  z  z  d||z  z   |z  z  S r  rq   rg  hyp2f1)r?   rk   r   r   r  r  normalization_constants          r0   rl   zgausshyper_gen._pdf  sm    !#A1aQ1K!K))ABK726QW:MM19q.! 	"r2   c                     t        j                  ||z   |      t        j                  ||      z  }t        j                  |||z   ||z   |z   |       }t        j                  ||||z   |       }||z  |z  S rH   r)  )	r?   r\   r   r   r  r  r  r  r  s	            r0   r   zgausshyper_gen._munp  sn    ggac1o1-ii1Q3!Ar*ii1acA2&3w}r2   N)r}   r~   r   r   r]   re   rl   r   r   r2   r0   r$  r$    s    @7 "
r2   r$  
gausshyperc                   `    e Zd ZdZej
                  Zd Zd Zd Z	d Z
d Zd Zd Zdd	Zd
 Zy)invgamma_gena_  An inverted gamma continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `invgamma` is:

    .. math::

        f(x, a) = \frac{x^{-a-1}}{\Gamma(a)} \exp(-\frac{1}{x})

    for :math:`x >= 0`, :math:`a > 0`. :math:`\Gamma` is the gamma function
    (`scipy.special.gamma`).

    `invgamma` takes ``a`` as a shape parameter for :math:`a`.

    `invgamma` is a special case of `gengamma` with ``c=-1``, and it is a
    different parameterization of the scaled inverse chi-squared distribution.
    Specifically, if the scaled inverse chi-squared distribution is
    parameterized with degrees of freedom :math:`\nu` and scaling parameter
    :math:`\tau^2`, then it can be modeled using `invgamma` with
    ``a=`` :math:`\nu/2` and ``scale=`` :math:`\nu \tau^2/2`.

    %(after_notes)s

    %(example)s

    c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zinvgamma_gen._shape_info1  r  r2   c                 L    t        j                  | j                  ||            S rH   r  r  s      r0   rl   zinvgamma_gen._pdf4  rx  r2   c                 r    |dz    t        j                  |      z  t        j                  |      z
  d|z  z
  S rz  rJ   r   rq   r  r  s      r0   r   zinvgamma_gen._logpdf8  s1    1vq	!BJJqM1CE99r2   c                 4    t        j                  |d|z        S r  r  r  s      r0   ro   zinvgamma_gen._cdf;  s    ||AsQw''r2   c                 4    dt        j                  ||      z  S r  r  r  s      r0   rx   zinvgamma_gen._ppf>  s    R__Q***r2   c                 4    t        j                  |d|z        S r  r  r  s      r0   rs   zinvgamma_gen._sfA  s    {{1cAg&&r2   c                 4    dt        j                  ||      z  S r  r  r  s      r0   r{   zinvgamma_gen._isfD  s    R^^Aq)))r2   c                 0   t        |dkD  |fd t        j                        }t        |dkD  |fd t        j                        }d\  }}d|v r!t        |dkD  |fd t        j                        }d	|v r!t        |d
kD  |fd t        j                        }||||fS )Nr   c                     d| dz
  z  S r  r   r   s    r0   r  z%invgamma_gen._stats.<locals>.<lambda>H  s    rQV} r2   rO   c                 $    d| dz
  dz  z  | dz
  z  S )Nr   rO   r   r   r   s    r0   r  z%invgamma_gen._stats.<locals>.<lambda>I  s    rQVaK/?1r6/J r2   r   r  r  c                 D    dt        j                  | dz
        z  | dz
  z  S )NrA  r   r;  r  r   s    r0   r  z%invgamma_gen._stats.<locals>.<lambda>P  s     "rwwq2v.!b&9 r2   r
  r  c                 0    dd| z  dz
  z  | dz
  z  | dz
  z  S )Nr:  r  g      &@r;  rA  r   r   s    r0   r  z%invgamma_gen._stats.<locals>.<lambda>T  s%    "Q-R8AFC r2   ru  )r?   r   r  r  r  r>  r?  s          r0   r   zinvgamma_gen._statsG  s    At%<bffEAt%J  B'>At9266CB '>AtCRVVMB 2r2~r2   c                 8    d }d }t        |dk\  |f||      }|S )Nc                 n    | | dz   t        j                  |       z  z
  t        j                  |       z   }|S r  r  r   r  s     r0   r  z&invgamma_gen._entropy.<locals>.regularX  s/    QWq	))BJJqM9AHr2   c                     ddt        j                  |       z  z
  t        j                  d      z   t        j                  t         j                        z   dz  d| dz  z  z   | dz  dz  z   | dz  d	z  z
  | d
z  dz  z
  }|S )Nr   r  rO   UUUUUU?r3  r  r  r  r  r  r  r   r?  s     r0   r$  z)invgamma_gen._entropy.<locals>.asymptotic\  s     aq	k/BFF1I-ruu=q@q#v: !3r	*,-sF2I6893s
CAHr2   r%  r&  r  )r?   r   r  r$  r  s        r0   r   zinvgamma_gen._entropyW  s)    		 qCx!@r2   Nmvsk)r}   r~   r   r   r   r  r  re   rl   r   ro   rx   rs   r{   r   r   r   r2   r0   r/  r/    sB    : "44ME*:(+'* r2   r/  invgammac                        e Zd ZdZej
                  Zd ZddZd Z	d Z
d Zd Zd Zd	 Z fd
Z fdZd Z ee       fd       Zd Z xZS )invgauss_gena`  An inverse Gaussian continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `invgauss` is:

    .. math::

        f(x; \mu) = \frac{1}{\sqrt{2 \pi x^3}}
                    \exp\left(-\frac{(x-\mu)^2}{2 \mu^2 x}\right)

    for :math:`x \ge 0` and :math:`\mu > 0`.

    `invgauss` takes ``mu`` as a shape parameter for :math:`\mu`.

    %(after_notes)s

    A common shape-scale parameterization of the inverse Gaussian distribution
    has density

    .. math::

        f(x; \nu, \lambda) = \sqrt{\frac{\lambda}{2 \pi x^3}}
                    \exp\left( -\frac{\lambda(x-\nu)^2}{2 \nu^2 x}\right)

    Using ``nu`` for :math:`\nu` and ``lam`` for :math:`\lambda`, this
    parameterization is equivalent to the one above with ``mu = nu/lam``,
    ``loc = 0``, and ``scale = lam``.

    %(example)s

    c                 @    t        dddt        j                  fd      gS Nr<  Fr   r  rb   rd   s    r0   re   zinvgauss_gen._shape_info  r  r2   c                 *    |j                  |d|      S Nr   r  waldr?   r<  r   r   s       r0   r   zinvgauss_gen._rvs  s      St 44r2   c                     dt        j                  dt         j                  z  |dz  z        z  t        j                  dd|z  z  ||z
  |z  dz  z        z  S )Nr   rO   r;  r3  r  r?   rk   r<  s      r0   rl   zinvgauss_gen._pdf  sO     2771RUU71c6>**266$!*qtRi!^2K+LLLr2   c                     dt        j                  dt         j                  z        z  dt        j                  |      z  z
  ||z
  |z  dz  d|z  z  z
  S )Nr  rO   r  r   rO  s      r0   r   zinvgauss_gen._logpdf  sH    BFF1RUU7O#c"&&)m3"by1nac6JJJr2   c                     dt        j                  |      z  }t        |||z  dz
  z        }d|z  t        | ||z  dz   z        z   }|t        j                  t        j                  ||z
              z   S r*  )rJ   r   r   r  r   r?   rk   r<  r  r   r   s         r0   r   zinvgauss_gen._logcdf  sl    "''!*nR1-.F\3$1r6Q,"788288BFF1q5M***r2   c                     dt        j                  |      z  }t        |||z  dz
  z        }d|z  t        | ||z   z  |z        z   }|t        j                  t        j
                  ||z
               z   S r*  )rJ   r   r   r   r  r   rR  s         r0   r   zinvgauss_gen._logsf  sn    "''!*nB!|,-F\3$!b&/B"677288RVVAE]N+++r2   c                 L    t        j                  | j                  ||            S rH   r-  rO  s      r0   rs   zinvgauss_gen._sf  s    vvdkk!R())r2   c                 L    t        j                  | j                  ||            S rH   r  rO  s      r0   ro   zinvgauss_gen._cdf      vvdll1b)**r2   c                 p   t        j                  ddd      5  t        j                  ||      \  }}t        j                  ||d      }|dkD  }t        j
                  d||   z
  ||   d      ||<   t        j                  |      }t        | !  ||   ||         ||<   d d d        |S # 1 sw Y   S xY wNr2  )r4  rk  invalidr   r   )	rJ   r5  r  rl  _invgauss_ppf_invgauss_isfrX  r;   rx   )r?   rk   r<  ppfi_wti_nanr  s         r0   rx   zinvgauss_gen._ppf      [[xJ 	;''2.EAr&&q"a0Cs7D,,QqwY4!DCIHHSMEah5	:CJ	; 
	; 
   BB++B5c                 p   t        j                  ddd      5  t        j                  ||      \  }}t        j                  ||d      }|dkD  }t        j
                  d||   z
  ||   d      ||<   t        j                  |      }t        | !  ||   ||         ||<   d d d        |S # 1 sw Y   S xY wrX  )	rJ   r5  r  rl  r[  rZ  rX  r;   r{   )r?   rk   r<  isfr]  r^  r  s         r0   r{   zinvgauss_gen._isf  r_  r`  c                 F    ||dz  dt        j                  |      z  d|z  fS )Nr;  r  r  r  )r?   r<  s     r0   r   zinvgauss_gen._stats  s%    2s7AbggbkM2b500r2   c                    |j                  dd      }t        |t              s%t        |       t        k(  s|j                         dk(  rt        	|   |g|i |S t        | |||      \  }}}}	 ||t        	|   |g|i |S t        j                  ||z
  dk        rt        ddt        j                        ||z
  }t        j                  |      }|*t        |      t        j                  |dz  |dz  z
        z  }||z  }|||fS )Nr,   r5   r6   r   invgaussr  r  )r7   r9   r%   r<   wald_genr8   r;   r=   r  rJ   r  rD  rc   r   r  r  )
r?   r@   rA   r/   r,   fshape_sr   r   fshape_nr  s
            r0   r=   zinvgauss_gen.fit  s   (E*t\*d4jH.D<<>T)7;t3d3d33'B4CG(O$hf	 <8/7;t3d3d33VVD4K!O$z"&&AA$;Dwwt}H~TbffTRZ(b.-H&IJ&(Hv%%r2   c                     dt        j                  dt         j                  z        z   dt        j                  |      z  z   }d|z  }t        j                  j                  |      |z  }d|z  d|z  z
  S )zV
        Ref.: https://moser-isi.ethz.ch/docs/papers/smos-2012-10.pdf (eq. 9)
        r   rO   r  r   r  )rJ   r   r   rq   r  r  )r?   r<  r   r  r   s        r0   r   zinvgauss_gen._entropy  sg     BEE	""Q^3 bDJJ##A&q(Qwq  r2   r   )r}   r~   r   r   r   r  r  re   r   rl   r   r   r   rs   ro   rx   r{   r   r
   r=   r   r  r  s   @r0   rF  rF  j  so    !D "44MF5M
K+,*+1 M*& +&B!r2   rF  re  c                   N    e Zd ZdZd Zd Zd Zd Zd Zd Z	dd	Z
d
 Zd Zd Zy)geninvgauss_genaX  A Generalized Inverse Gaussian continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `geninvgauss` is:

    .. math::

        f(x, p, b) = x^{p-1} \exp(-b (x + 1/x) / 2) / (2 K_p(b))

    where `x > 0`, `p` is a real number and `b > 0`\([1]_).
    :math:`K_p` is the modified Bessel function of second kind of order `p`
    (`scipy.special.kv`).

    %(after_notes)s

    The inverse Gaussian distribution `stats.invgauss(mu)` is a special case of
    `geninvgauss` with `p = -1/2`, `b = 1 / mu` and `scale = mu`.

    Generating random variates is challenging for this distribution. The
    implementation is based on [2]_.

    References
    ----------
    .. [1] O. Barndorff-Nielsen, P. Blaesild, C. Halgreen, "First hitting time
       models for the generalized inverse gaussian distribution",
       Stochastic Processes and their Applications 7, pp. 49--54, 1978.

    .. [2] W. Hoermann and J. Leydold, "Generating generalized inverse Gaussian
       random variates", Statistics and Computing, 24(4), p. 547--557, 2014.

    %(example)s

    c                     ||k(  |dkD  z  S r  r   r?   r  r   s      r0   r]   zgeninvgauss_gen._argcheck"  s    Q1q5!!r2   c                     t        ddt        j                   t        j                  fd      }t        dddt        j                  fd      }||gS )Nr  Fr  r   r   rb   )r?   r:  rc  s      r0   re   zgeninvgauss_gen._shape_info%  B    UbffWbff$5~FUQK@Bxr2   c                     d }t        j                  |t         j                  g      } ||||      }t        j                  |      j	                         rd}t        j                  |t        d       |S )Nc                 0    t        j                  | ||      S rH   )r   geninvgauss_logpdfrk   r  r   s      r0   logpdf_singlez.geninvgauss_gen._logpdf.<locals>.logpdf_single.  s    ,,Q155r2   rH  zjInfinite values encountered in scipy.special.kve(p, b). Values replaced by NaN to avoid incorrect results.r  r  )rJ   rB  rL  rX  r  r  r  r  )r?   rk   r  r   rt  r  rS   s          r0   r   zgeninvgauss_gen._logpdf*  s\    	6 ]BJJ<H!Q"88A;??HCMM#~!<r2   c                 N    t        j                  | j                  |||            S rH   r  r?   rk   r  r   s       r0   rl   zgeninvgauss_gen._pdf:  r  r2   c                      | j                   | \  }fd}t        j                  |t        j                  g      } ||g| S )Nc                     |\  }}t        j                  ||gt              j                  j	                  t        j
                        }t        j                  t        d|      }t        j                  ||       d   S )N_geninvgauss_pdfr   )rJ   rQ  rA  rR  rS  rT  r   rU  r   r   rW  )rk   rA   r  r   r[  r\  r  s         r0   _cdf_singlez)geninvgauss_gen._cdf.<locals>._cdf_singleA  sg    DAq!Q/66>>vOI"..v7I/8:C >>#r1-a00r2   rH  )r   rJ   rB  rL  )r?   rk   rA   r  rz  r  s        @r0   ro   zgeninvgauss_gen._cdf>  sF    """D)B	1 ll;

|D1$t$$r2   c                 J    t        |dkD  |||fd t        j                         S )Nr   c                 V    |dz
  t        j                  |       z  || d| z  z   z  dz  z
  S r*  r+  rs  s      r0   r  z.geninvgauss_gen._logquasipdf.<locals>.<lambda>P  s,    1q5"&&)*;aQqSk!m*K r2   r  rv  s       r0   _logquasipdfzgeninvgauss_gen._logquasipdfM  s)    !a%!QK66'# 	#r2   Nc                 X  	
 t        j                  |      r+t        j                  |      r| j                  ||||      }nR|j                  dk(  rA|j                  dk(  r2| j                  |j	                         |j	                         ||      }nt        j
                  ||      \  }}t        |j                  |      \  }	t        t        j                  |            }t        j                  |      }t        j                  ||gdgdgdgg      

j                  srt        	
fdt        t        |       d      D              }| j                  
d   
d   ||      j!                  |      ||<   
j#                          
j                  sr|dk(  r|j	                         }|S )Nr   multi_indexreadonlyflagsop_flagsc              3   \   K   | ]#  }|   sj                   |   n
t        d        % y wrH   r  slicerj  r  bcits     r0   rl  z'geninvgauss_gen._rvs.<locals>.<genexpr>  1      ; ! 79eR^^A.tL ;   ),r   r   )rJ   isscalar_rvs_scalarr   rG  r  r   rn  r  r  emptynditerfinishedtupler  r  rE  iternext)r?   r  r   r   r   r  shp
numsamplesidxr  r  s            @@r0   r   zgeninvgauss_gen._rvsS  si    ;;q>bkk!n""1a|<CVVq[QVVq[""1668QVVXt\JC &&q!,DAq #177D1GC RWWS\*J ((4.CAq6"/&0\J<$@BB kk  ;%*CI:q%9; ;++BqE2a5*,8::A'#, C kk  2:((*C
r2   c           	         3 d}|sd}dk  r d} j                        }d}dk\  sdkD  rd}n0t        ddt        j                  dz
        z  dz        k\  rd}nd}t	        t        j
                  |            }	t        j                  |	      }
t        j                  |
      }d}|rrqddz   z  z  |z
  }d|z  dz
  z  z  dz
  }||dz  dz  z
  }d|dz  z  d	z  ||z  dz  z
  |z   }t        j                  | t        j                  d
|dz  z        z  dz        }t        j                  d|z  dz         }|t        j                  |dz  t        j                  dz  z         z  |dz  z
  }| t        j                  |dz        z  |dz  z
  } j                  |      3 j                  |      3z
  } j                  |      3z
  }||z
  t        j                  d|z        z  }||z
  t        j                  d|z        z  }d}3 fd}|}nt        j                  d j                  |      z        }dz   t        j                  dz   dz  dz  z         z   z  }d}|t        j                  d j                  |      z        z  }d} fd}||k\  rt        d      |dk  rt        d      d}||
k  rr|
|z
  }||j                  |      z  }|j                  |      } |||z
  | z  z   } | |z  |z   }!dt        j                  |      z   ||!      k  }"t        j                   |"      }#|#dkD  r|!|"   ||||#z    ||#z  }|dk(  r||
z  dk\  rd||
z   d}$t#        |$      |dz  }||
k  rndz
  z  }%t        j$                  |%dz  f      }&t        j                   j                  |            }'|'|%z  }(|%dz  k  rOt        j                         })dkD  r|)dz  z  |%z  z
  z  z  }*n$|)t        j                  ddz  z        z  }*nd\  })}*|&dz
  z  }+d|+z  t        j                  |& z  dz        z  z  },|(|*z   |,z   }-||
k  r|
|z
  }t        j                  |      t        j                  |      }!}.|j                  |      }|-|j                  |      z  } | |(k  }/t        j&                  |/      | |(|*z   k  z  }0t        j&                  |/|0z        }1|%| |/   z  |(z  |!|/<   |'|.|/<   dkD  r|%z  | |0   |(z
  z  |)z  z   dz  z  |!|0<   n7t        j                  | |0   |(z
  t        j                        z        z  |!|0<   |)|!|0   dz
  z  z  |.|0<   t        j                  |& z  dz        | |1   |(z
  |*z
  z  d|+z  z  z
  }2dz  t        j                  |2      z  |!|1<   |+t        j                  |!|1    z  dz        z  |.|1<   t        j                  ||.z         j                  |!      k  }"t!        |"      }#|#dkD  r|!|"   ||||#z    ||#z  }||
k  rt        j(                  ||	      }!|rd|!z  }!|!S )NFr   r   Tr   rO   r  r	     ir  c                 0    j                  |       z
  S rH   r}  )rk   r   lmr  r?   s    r0   logqpdfz,geninvgauss_gen._rvs_scalar.<locals>.logqpdf  s    ,,Q15::r2   c                 *    j                  |       S rH   r  )rk   r   r  r?   s    r0   r  z,geninvgauss_gen._rvs_scalar.<locals>.logqpdf  s    ,,Q155r2   zvmin must be smaller than vmax.zumax must be positive.r  iP  z2Not a single random variate could be generated in zH attempts. Sampling does not appear to work for the provided parameters.)r   r   )_moder@  rJ   r   r  
atleast_1dr  zerosarccosr  r   r}  r   r   r  r   r  r  rY  logical_notrE  )4r?   r  r   r  r   
invert_resr  
ratio_unif
mode_shiftsize1dNrk   	simulateda2a1p1q1phirU  root1root2d1d2vminvmaxumaxr  r  xplusr  r
  r  r  r  accept
num_acceptrS   r  xsk1A1k2A2k3A3r!  r  cond1cond2cond3r  r  s4   ```                                                @r0   r  zgeninvgauss_gen._rvs_scalar  s    
Jq5AJJJq! 
6QUJ#c1rwwq1u~-122J J r}}Z01GGFOHHQK	1q5\A%)Ua!e_q(1,"a%!)^QY^b2gk1A5iibggcBEk&: :Q >?ggb2gk**RVVC!Gbeeai$78826AbffS1Wo-Q6 &&q!Q/&&ua3b8&&ua3b8 	RVVC"H%55	RVVC"H%55;  vvc$"3"3Aq!"<<=a%277AEA:1+<#==q@rvvcD,=,=eQ,J&JKK6 t| !BCCqy !9::Aa-	M<//Q/77 ((a(0D4K1,,!eaiBFF1I+5VVF^
><?KAiZ!79+IN1!!"1 &??C 's++Q' a-, a!eBQU$B))!Q23BbBAEzVVQBZq5AzBE12Q6BbffQAX..BBa!eBR"&&"q1--1BR"A a-	M!bhhqk3 ((a(0,,!,44Ru-b2g>uu}5!E(]R/E
%q5"$a%1U8b=A*=*B"Ba!e!LCJ!"RVVQuX]bffQi,G%H!HCJE
QU 33%FFB37Q;'!qx"}r/A*Ba"f*MM!VbffQi/E
E
{Q': ;;%&&Q-4+<+<S!Q+GG [
><?KAiZ!79+I7 a-: jjF#c'C
r2   c                     |dk  r*|t        j                  |dz
  dz  |dz  z         dz   |z
  z  S t        j                  d|z
  dz  |dz  z         d|z
  z
  |z  S r*  r  rm  s      r0   r  zgeninvgauss_gen._mode#  sf    q5Q
QT 12Q6:;;GGQUQJA-.!a%8A==r2   c                    t        j                  ||z   |      }t        j                  ||      }t        j                  |      t        j                  |      z  }|j	                         red}t        j                  |t        d       t        j                  |t        j                  t        j                        }||    ||    z  || <   |S ||z  }|S )NzInfinite values encountered in the moment calculation involving scipy.special.kve. Values replaced by NaN to avoid incorrect results.r  r  dtype)rq   kverJ   rP   r  r  r  r  	full_liker  rL  )	r?   r\   r  r   r  denominf_valsrS   r  s	            r0   r   zgeninvgauss_gen._munp*  s    ffQUAq!88C=288E?2<<>.C MM#~!<S"&&

;Ay>E8),<<AxiL  eAr2   r   )r}   r~   r   r   r]   re   r   rl   ro   r}  r   r  r  r   r   r2   r0   rk  rk    s=    #H"
 -%#5nWr>r2   rk  re  c                   f     e Zd ZdZej
                  Zd Zd Z fdZ	d Z
d Zd Zd
dZd	 Z xZS )norminvgauss_gena  A Normal Inverse Gaussian continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `norminvgauss` is:

    .. math::

        f(x, a, b) = \frac{a \, K_1(a \sqrt{1 + x^2})}{\pi \sqrt{1 + x^2}} \,
                     \exp(\sqrt{a^2 - b^2} + b x)

    where :math:`x` is a real number, the parameter :math:`a` is the tail
    heaviness and :math:`b` is the asymmetry parameter satisfying
    :math:`a > 0` and :math:`|b| <= a`.
    :math:`K_1` is the modified Bessel function of second kind
    (`scipy.special.k1`).

    %(after_notes)s

    A normal inverse Gaussian random variable `Y` with parameters `a` and `b`
    can be expressed as a normal mean-variance mixture:
    `Y = b * V + sqrt(V) * X` where `X` is `norm(0,1)` and `V` is
    `invgauss(mu=1/sqrt(a**2 - b**2))`. This representation is used
    to generate random variates.

    Another common parametrization of the distribution (see Equation 2.1 in
    [2]_) is given by the following expression of the pdf:

    .. math::

        g(x, \alpha, \beta, \delta, \mu) =
        \frac{\alpha\delta K_1\left(\alpha\sqrt{\delta^2 + (x - \mu)^2}\right)}
        {\pi \sqrt{\delta^2 + (x - \mu)^2}} \,
        e^{\delta \sqrt{\alpha^2 - \beta^2} + \beta (x - \mu)}

    In SciPy, this corresponds to
    `a = alpha * delta, b = beta * delta, loc = mu, scale=delta`.

    References
    ----------
    .. [1] O. Barndorff-Nielsen, "Hyperbolic Distributions and Distributions on
           Hyperbolae", Scandinavian Journal of Statistics, Vol. 5(3),
           pp. 151-157, 1978.

    .. [2] O. Barndorff-Nielsen, "Normal Inverse Gaussian Distributions and
           Stochastic Volatility Modelling", Scandinavian Journal of
           Statistics, Vol. 24, pp. 1-13, 1997.

    %(example)s

    c                 >    |dkD  t        j                  |      |k  z  S r  )rJ   absoluter  s      r0   r]   znorminvgauss_gen._argchecku  s    A"++a.1,--r2   c                     t        dddt        j                  fd      }t        ddt        j                   t        j                  fd      }||gS r`  rb   ra  s      r0   re   znorminvgauss_gen._shape_infox  r  r2   c                 &    t         |   |d      S )N)r   r   rF  r  r  s     r0   r  znorminvgauss_gen._fitstart}  s     w H 55r2   c                    t        j                  |dz  |dz  z
        }|t         j                  z  }t        j                  d|      }|t	        j
                  ||z        z  t        j                  ||z  ||z  z
  |z         z  |z  S r  )rJ   r   r   hypotrq   k1er   )r?   rk   r   r   r  fac1sqs          r0   rl   znorminvgauss_gen._pdf  ss    1q!t$255yXXa^bffQVn$rvvacAbDj5.@'AABFFr2   c           
         t        j                  |      r6t        j                  | j                  |t         j
                  ||f      d   S t        j                  |      }t        j                  |      }g }t        |||      D ]K  \  }}}|j                  t        j                  | j                  |t         j
                  ||f      d          M t        j                  |      S )NrF  r   )
rJ   r  r   rW  rl   rc   r  rx  appendrQ  )r?   rk   r   r   resultr  a0rk  s           r0   rs   znorminvgauss_gen._sf  s    ;;q>>>$))QaVDQGGa Aa AF #Aq! @RinnTYYBFF35r(<<=? @@ 88F##r2   c                       fd}t        j                  |      r
 ||||      S g }t        |||      D ]  \  }}}|j                   ||||             ! t        j                  |      S )Nc                 t   
fd}
j                  ||      } |||||       }|dk(  r|S |dkD  r1d}|}||z   } |||||       dkD  rJd|z  }||z   } |||||       dkD  rn0d}|}||z
  } |||||       dk  rd|z  }||z
  } |||||       dk  rt        j                  |||||| f
j                        }	|	S )Nc                 0    j                  | ||      |z
  S rH   rs   )rk   r   r   rw   r?   s       r0   eqz6norminvgauss_gen._isf.<locals>._isf_scalar.<locals>.eq  s    xx1a(1,,r2   r   r   rO   )rA   r  )r   r   r  r  )rw   r   r   r  xmemdeltaleftrightr  r?   s             r0   _isf_scalarz*norminvgauss_gen._isf.<locals>._isf_scalar  s   - 1aBB1aBQw	AvU
1a(1,eGEJE 1a(1,
 Ezq!Q'!+eGE:D q!Q'!+ __RuAq!9*.))5FMr2   )rJ   r  rx  r  rQ  )	r?   rw   r   r   r  r  q0r  rk  s	   `        r0   r{   znorminvgauss_gen._isf  sk    	B ;;q>q!Q''F #Aq! 7Rk"b"56788F##r2   c                     t        j                  |dz  |dz  z
        }t        j                  d|z  ||      }||z  t        j                  |      t        j                  ||      z  z   S )NrO   r   )r<  r   r   r  )rJ   r   re  r  r   )r?   r   r   r   r   r  igs          r0   r   znorminvgauss_gen._rvs  so     1q!t$\\QuW4l\K2vdhhD<H '/ 'J J J 	Jr2   c                     t        j                  |dz  |dz  z
        }||z  }|dz  |dz  z  }d|z  |t        j                  |      z  z  }ddd|dz  z  |dz  z  z   z  |z  }||||fS )NrO   r  r;  r   r  r  )r?   r   r   r  r   varianceskewnesskurtosiss           r0   r   znorminvgauss_gen._stats  s    1q!t$5ya4%(?7a"''%.01!a!Q$hAo-.6Xx11r2   r   )r}   r~   r   r   r   r  r  r]   re   r  rl   rs   r{   r   r   r  r  s   @r0   r  r  =  sA    4j "44M.
6
G$($TJ2r2   r  norminvgaussc                   l     e Zd ZdZej
                  Zd Zd Zd Z	d Z
d Zd Zd Zd	 Zd fd
	Z xZS )invweibull_genu  An inverted Weibull continuous random variable.

    This distribution is also known as the Fréchet distribution or the
    type II extreme value distribution.

    %(before_notes)s

    Notes
    -----
    The probability density function for `invweibull` is:

    .. math::

        f(x, c) = c x^{-c-1} \exp(-x^{-c})

    for :math:`x > 0`, :math:`c > 0`.

    `invweibull` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    References
    ----------
    F.R.S. de Gusmao, E.M.M Ortega and G.M. Cordeiro, "The generalized inverse
    Weibull distribution", Stat. Papers, vol. 52, pp. 591-619, 2011.

    %(example)s

    c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zinvweibull_gen._shape_info  r  r2   c                     t        j                  || dz
        }t        j                  ||       }t        j                  |       }||z  |z  S r  rJ   r  r   )r?   rk   r  xc1xc2s        r0   rl   zinvweibull_gen._pdf  sF    hhq1"s(#hhq1"offcTl3w}r2   c                 \    t        j                  ||       }t        j                  |       S rH   r  )r?   rk   r  r  s       r0   ro   zinvweibull_gen._cdf  s#    hhq1"ovvsd|r2   c                 8    t        j                  || z          S rH   )rJ   r  r  s      r0   rs   zinvweibull_gen._sf  s    !aR%   r2   c                 \    t        j                  t        j                  |       d|z        S r6  )rJ   r  r   r  s      r0   rx   zinvweibull_gen._ppf  s!    xx
DF++r2   c                 <    t        j                  |        d|z  z  S Nr  r0  r  s      r0   r{   zinvweibull_gen._isf  s    1"A&&r2   c                 8    t        j                  d||z  z
        S rX   r"  rp  s      r0   r   zinvweibull_gen._munp
  s    xxAE	""r2   c                 T    dt         z   t         |z  z   t        j                  |      z
  S rX   r  rr  s     r0   r   zinvweibull_gen._entropy  s"    x&1*$rvvay00r2   c                 2    |dn|}t         |   ||      S )N)r   rF  r  )r?   r@   rA   r  s      r0   r  zinvweibull_gen._fitstart  s#    v4w D 11r2   rH   )r}   r~   r   r   r   r  r  re   rl   ro   rs   rx   r{   r   r   r  r  r  s   @r0   r  r    sH    : "44ME!,'#12 2r2   r  
invweibullc                   6    e Zd ZdZd Zd Zd	dZd Zd Zd Z	y)
jf_skew_t_gena  Jones and Faddy skew-t distribution.

    %(before_notes)s

    Notes
    -----
    The probability density function for `jf_skew_t` is:

    .. math::

        f(x; a, b) = C_{a,b}^{-1}
                    \left(1+\frac{x}{\left(a+b+x^2\right)^{1/2}}\right)^{a+1/2}
                    \left(1-\frac{x}{\left(a+b+x^2\right)^{1/2}}\right)^{b+1/2}

    for real numbers :math:`a>0` and :math:`b>0`, where
    :math:`C_{a,b} = 2^{a+b-1}B(a,b)(a+b)^{1/2}`, and :math:`B` denotes the
    beta function (`scipy.special.beta`).

    When :math:`a<b`, the distribution is negatively skewed, and when
    :math:`a>b`, the distribution is positively skewed. If :math:`a=b`, then
    we recover the `t` distribution with :math:`2a` degrees of freedom.

    `jf_skew_t` takes :math:`a` and :math:`b` as shape parameters.

    %(after_notes)s

    References
    ----------
    .. [1] M.C. Jones and M.J. Faddy. "A skew extension of the t distribution,
           with applications" *Journal of the Royal Statistical Society*.
           Series B (Statistical Methodology) 65, no. 1 (2003): 159-174.
           :doi:`10.1111/1467-9868.00378`

    %(example)s

    c                     t        dddt        j                  fd      }t        dddt        j                  fd      }||gS r`  rb   ra  s      r0   re   zjf_skew_t_gen._shape_info>  rd  r2   c                 0   d||z   dz
  z  t        j                  ||      z  t        j                  ||z         z  }d|t        j                  ||z   |dz  z         z  z   |dz   z  }d|t        j                  ||z   |dz  z         z  z
  |dz   z  }||z  |z  S NrO   r   r   )rq   rg  rJ   r   )r?   rk   r   r   r  r  r  s          r0   rl   zjf_skew_t_gen._pdfC  s    !a%!)rwwq!},rwwq1u~=!bgga!ea1fn---1s7;!bgga!ea1fn---1s7;Bw{r2   Nc                     |j                  |||      }d|z  dz
  t        j                  ||z         z  }dt        j                  |d|z
  z        z  }||z  S r  )rg  rJ   r   )r?   r   r   r   r   r  r  d3s           r0   r   zjf_skew_t_gen._rvsI  sX    q!T*"fqjBGGAEN*q2v''Bwr2   c                 ~    d|t        j                  ||z   |dz  z         z  z   dz  }t        j                  |||      S Nr   rO   r   )rJ   r   rq   betainc)r?   rk   r   r   r^  s        r0   ro   zjf_skew_t_gen._cdfO  s>    RWWQUQ!V^,,,3zz!Q""r2   c                     t         j                  |||      }d|z  dz
  t        j                  ||z         z  }dt        j                  |d|z
  z        z  }||z  S r  )rg  r\  rJ   r   )r?   rw   r   r   r  r  r  s          r0   rx   zjf_skew_t_gen._ppfS  sV    XXaA"fqjBGGAEN*q2v''Bwr2   c                     d }|d|z  kD  |d|z  kD  z  |dk\  z  }t        ||||ft        j                  |t        j                  g      t        j                        S )zReturns the n-th moment(s) where all the following hold:

        - n >= 0
        - a > n / 2
        - b > n / 2

        The result is np.nan in all other cases.
        c                 n   ||z   d| z  z  }d| z  t        j                  ||      z  }t        j                  | dz         }t        j                  |dz  dkD  dd      }t        j                  |d| z  z   |z
  |d| z  z
  |z         }t        j
                  | |      |z  |z  }||z  |j                         z  S )zgComputes E[T^(n_k)] where T is skew-t distributed with
            parameters a_k and b_k.
            r   rO   r   r   r  )rq   rg  rJ   rD  rF  ry  r  )	n_ka_kb_kr  r  indicesr8  r  	sum_termss	            r0   
nth_momentz'jf_skew_t_gen._munp.<locals>.nth_momentb  s     9#),CHrwwsC00Eiia(G((7Q;?B2CcCi'13s?W3LMAW-3a7I;00r2   r   r   rH  r   rJ   rB  rL  r  )r?   r\   r   r   r  nth_moment_valids         r0   r   zjf_skew_t_gen._munpY  sa    	1 aKAaK8AFC1ILLRZZL9FF	
 	
r2   r   )
r}   r~   r   r   re   rl   r   ro   rx   r   r   r2   r0   r  r    s&    #H
#
r2   r  	jf_skew_tc                   R    e Zd ZdZej
                  Zd Zd Zd Z	d Z
d Zd Zd Zy	)
johnsonsb_gena!  A Johnson SB continuous random variable.

    %(before_notes)s

    See Also
    --------
    johnsonsu

    Notes
    -----
    The probability density function for `johnsonsb` is:

    .. math::

        f(x, a, b) = \frac{b}{x(1-x)}  \phi(a + b \log \frac{x}{1-x} )

    where :math:`x`, :math:`a`, and :math:`b` are real scalars; :math:`b > 0`
    and :math:`x \in [0,1]`.  :math:`\phi` is the pdf of the normal
    distribution.

    `johnsonsb` takes :math:`a` and :math:`b` as shape parameters.

    %(after_notes)s

    %(example)s

    c                     |dkD  ||k(  z  S r  r   r  s      r0   r]   zjohnsonsb_gen._argcheck  r	  r2   c                     t        ddt        j                   t        j                  fd      }t        dddt        j                  fd      }||gS Nr   Fr  r   r   rb   ra  s      r0   re   zjohnsonsb_gen._shape_info  ro  r2   c                 l    t        ||t        j                  |      z  z         }|dz  |d|z
  z  z  |z  S r   )r   rq   r  )r?   rk   r   r   trms        r0   rl   zjohnsonsb_gen._pdf  s8    AbhhqkM)*ua1gs""r2   c                 J    t        ||t        j                  |      z  z         S rH   )r   rq   r  rn  s       r0   ro   zjohnsonsb_gen._cdf  s    Qrxx{]*++r2   c                 P    t        j                  d|z  t        |      |z
  z        S r  )rq   r  r   r}  s       r0   rx   zjohnsonsb_gen._ppf  #    xxa9Q<!#3455r2   c                 J    t        ||t        j                  |      z  z         S rH   )r   rq   r  rn  s       r0   rs   zjohnsonsb_gen._sf  s    AbhhqkM)**r2   c                 P    t        j                  d|z  t        |      |z
  z        S r  )rq   r  r   r}  s       r0   r{   zjohnsonsb_gen._isf  r  r2   N)r}   r~   r   r   r   r  r  r]   re   rl   ro   rx   rs   r{   r   r2   r0   r  r  |  s7    6 "44M"
#
,6+6r2   r  	johnsonsbc                   B    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
dd	Zy
)johnsonsu_gena-  A Johnson SU continuous random variable.

    %(before_notes)s

    See Also
    --------
    johnsonsb

    Notes
    -----
    The probability density function for `johnsonsu` is:

    .. math::

        f(x, a, b) = \frac{b}{\sqrt{x^2 + 1}}
                     \phi(a + b \log(x + \sqrt{x^2 + 1}))

    where :math:`x`, :math:`a`, and :math:`b` are real scalars; :math:`b > 0`.
    :math:`\phi` is the pdf of the normal distribution.

    `johnsonsu` takes :math:`a` and :math:`b` as shape parameters.

    The first four central moments are calculated according to the formulas
    in [1]_.

    %(after_notes)s

    References
    ----------
    .. [1] Taylor Enterprises. "Johnson Family of Distributions".
       https://variation.com/wp-content/distribution_analyzer_help/hs126.htm

    %(example)s

    c                     |dkD  ||k(  z  S r  r   r  s      r0   r]   zjohnsonsu_gen._argcheck  r	  r2   c                     t        ddt        j                   t        j                  fd      }t        dddt        j                  fd      }||gS r  rb   ra  s      r0   re   zjohnsonsu_gen._shape_info  ro  r2   c                     ||z  }t        ||t        j                  |      z  z         }|dz  t        j                  |dz         z  |z  S r  )r   rJ   arcsinhr   )r?   rk   r   r   r  r  s         r0   rl   zjohnsonsu_gen._pdf  sI     qSA

1--.uRWWRV_$S((r2   c                 J    t        ||t        j                  |      z  z         S rH   )r   rJ   r#  rn  s       r0   ro   zjohnsonsu_gen._cdf  s    QA..//r2   c                 J    t        j                  t        |      |z
  |z        S rH   )rJ   sinhr   r}  s       r0   rx   zjohnsonsu_gen._ppf      ww	!q(A-..r2   c                 J    t        ||t        j                  |      z  z         S rH   )r   rJ   r#  rn  s       r0   rs   zjohnsonsu_gen._sf  s    A

1--..r2   c                 J    t        j                  t        |      |z
  |z        S rH   )rJ   r&  r   rn  s       r0   r{   zjohnsonsu_gen._isf  r'  r2   c                    d\  }}}}|dz  }t        j                  |      }	||z  }
d|v r|	dz   t        j                  |
      z  }d|v r7dt        j                  |      z  |	t        j
                  d|
z        z  dz   z  }d|v r|	dz  t        j                  |      dz  z  }d	t        j                  |
      z  }|	|	dz   z  t        j                  d	|
z        z  }t        j                  d      d|	t        j
                  d|
z        z  z   d
z  z  }| ||z   z  |z  }d|v rd	d|	z  z   }d|	dz  z  |	dz   z  t        j
                  d|
z        z  }|	dz  t        j
                  d|
z        z  }dd	|	dz  z  z   d|	d	z  z  z   |	dz  z   }dd|	t        j
                  d|
z        z  z   dz  z  }||z   ||z  z   |z  d	z
  }||||fS )NNNNNr  r  r   r  rO   r   r  r  r  r
  r  r  r  )rJ   r   r&  rq   r  r  r   )r?   r   r   r  r<  r=  r>  r?  bn2expbn2a_br  r  r  r  t4s                   r0   r   zjohnsonsu_gen._stats  s    1CRf!e'>#+,B'>bhhsm#VBGGAcEN%:Q%>?C'>bhhsmS00B2773<B6A:&37BGGAJ!frwwqu~&="=!EEER5(B'>QvXB619
+bggaen<BRWWQsU^+Ba	k!AfaiK/&!);Bq6"''!C%.00144Er'BrE/U*Q.B3Br2   Nr  )r}   r~   r   r   r]   re   rl   ro   rx   rs   r{   r   r   r2   r0   r  r    s0    "F"
)0///r2   r  	johnsonsuc                   r    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 Zd Ze eed      d               Zy)laplace_gena
  A Laplace continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `laplace` is

    .. math::

        f(x) = \frac{1}{2} \exp(-|x|)

    for a real number :math:`x`.

    %(after_notes)s

    %(example)s

    c                     g S rH   r   rd   s    r0   re   zlaplace_gen._shape_info+  r   r2   Nc                 *    |j                  dd|      S )Nr   r   r  )laplacer   s      r0   r   zlaplace_gen._rvs.  s    ##Aqt#44r2   c                 F    dt        j                  t        |             z  S r  )rJ   r   r  r   s     r0   rl   zlaplace_gen._pdf1  s    2663q6'?""r2   c           	          t        j                  d      5  t        j                  |dkD  ddt        j                  |       z  z
  dt        j                  |      z        cd d d        S # 1 sw Y   y xY w)Nr2  rj  r   r   r   )rJ   r5  rF  r   r   s     r0   ro   zlaplace_gen._cdf5  sZ    [[h' 	H88AE3RVVQBZ#7RVVAYG	H 	H 	Hs   A
A++A4c                 &    | j                  |       S rH   ro   r   s     r0   rs   zlaplace_gen._sf9  s    yy!}r2   c                     t        j                  |dkD  t        j                  dd|z
  z         t        j                  d|z              S r   rJ   rF  r   r   s     r0   rx   zlaplace_gen._ppf=  s8    xxC"&&AaC/!1266!A#;??r2   c                 &    | j                  |       S rH   rL  r   s     r0   r{   zlaplace_gen._isf@  s    		!}r2   c                      y)N)r   rO   r   r  r   rd   s    r0   r   zlaplace_gen._statsD  s    r2   c                 2    t        j                  d      dz   S r  r+  rd   s    r0   r   zlaplace_gen._entropyG  s    vvay{r2   z        This function uses explicit formulas for the maximum likelihood
        estimation of the Laplace distribution parameters, so the keyword
        arguments `loc`, `scale`, and `optimizer` are ignored.

r   c                     t        | |||      \  }}}|t        j                  |      }|7t        j                  t        j                  ||z
              t        |      z  }||fS rH   )r  rJ   medianr  r  r  )r?   r@   rA   r/   r   r   s         r0   r=   zlaplace_gen.fitJ  sf     9t9=tEdF <99T?D>ffRVVD4K01SY>FV|r2   r   )r}   r~   r   r   re   r   rl   ro   rs   rx   r{   r   r   rE   r	   r   r=   r   r2   r0   r2  r2    sa    &5#H@  6F G	G 
r2   r2  r5  c                   F    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zy)laplace_asymmetric_genu  An asymmetric Laplace continuous random variable.

    %(before_notes)s

    See Also
    --------
    laplace : Laplace distribution

    Notes
    -----
    The probability density function for `laplace_asymmetric` is

    .. math::

       f(x, \kappa) &= \frac{1}{\kappa+\kappa^{-1}}\exp(-x\kappa),\quad x\ge0\\
                    &= \frac{1}{\kappa+\kappa^{-1}}\exp(x/\kappa),\quad x<0\\

    for :math:`-\infty < x < \infty`, :math:`\kappa > 0`.

    `laplace_asymmetric` takes ``kappa`` as a shape parameter for
    :math:`\kappa`. For :math:`\kappa = 1`, it is identical to a
    Laplace distribution.

    %(after_notes)s

    Note that the scale parameter of some references is the reciprocal of
    SciPy's ``scale``. For example, :math:`\lambda = 1/2` in the
    parameterization of [1]_ is equivalent to ``scale = 2`` with
    `laplace_asymmetric`.

    References
    ----------
    .. [1] "Asymmetric Laplace distribution", Wikipedia
            https://en.wikipedia.org/wiki/Asymmetric_Laplace_distribution

    .. [2] Kozubowski TJ and Podgórski K. A Multivariate and
           Asymmetric Generalization of Laplace Distribution,
           Computational Statistics 15, 531--540 (2000).
           :doi:`10.1007/PL00022717`

    %(example)s

    c                 @    t        dddt        j                  fd      gS )NkappaFr   r  rb   rd   s    r0   re   z"laplace_asymmetric_gen._shape_info  s    7EArvv;GHHr2   c                 L    t        j                  | j                  ||            S rH   r  r?   rk   rD  s      r0   rl   zlaplace_asymmetric_gen._pdf  s    vvdll1e,--r2   c                     d|z  }|t        j                  |dk\  | |      z  }|t        j                  ||z         z  }|S r@  r;  )r?   rk   rD  kapinvrt  s        r0   r   zlaplace_asymmetric_gen._logpdf  sD    5"((16E6622rvveFl##
r2   c                     d|z  }||z   }t        j                  |dk\  dt        j                  | |z        ||z  z  z
  t        j                  ||z        ||z  z        S r@  rJ   rF  r   r?   rk   rD  rH  
kappkapinvs        r0   ro   zlaplace_asymmetric_gen._cdf  sf    56\
xxQBFFA2e8,fZ.?@@qx(%
*:;= 	=r2   c           	          d|z  }||z   }t        j                  |dk\  t        j                  | |z        ||z  z  dt        j                  ||z        ||z  z  z
        S r@  rJ  rK  s        r0   rs   zlaplace_asymmetric_gen._sf  sh    56\
xxQr%x(&*;<BFF1V8,eJ.>??A 	Ar2   c                     d|z  }||z   }t        j                  |||z  k\  t        j                  d|z
  |z  |z         |z  t        j                  ||z  |z        |z        S rX   r;  r?   rw   rD  rH  rL  s        r0   rx   zlaplace_asymmetric_gen._ppf  sm    56\
xxU:--Q
 25 899&@q|E1258: 	:r2   c                     d|z  }||z   }t        j                  |||z  k  t        j                  ||z  |z         |z  t        j                  d|z
  |z  |z        |z        S rX   r;  rO  s        r0   r{   zlaplace_asymmetric_gen._isf  so    56\
xxVJ..*U 233F:Az1%78>@ 	@r2   c                 `   d|z  }||z
  }||z  ||z  z   }ddt        j                  |d      z
  z  t        j                  dt        j                  |d      z   d      z  }ddt        j                  |d      z   z  t        j                  dt        j                  |d      z   d      z  }||||fS )	Nr   r   r  r  r  r:  r'  rO   r  )r?   rD  rH  mnr  r>  r?  s          r0   r   zlaplace_asymmetric_gen._stats  s    5e^VmeEk)!BHHUA&&'288E13E1Es(KK!BHHUA&&'288E13E1Eq(II3Br2   c                 >    dt        j                  |d|z  z         z   S rX   r+  r?   rD  s     r0   r   zlaplace_asymmetric_gen._entropy  s    266%%-(((r2   Nr  r   r2   r0   rB  rB  b  s8    *VI.=A:@)r2   rB  laplace_asymmetricc                    t        |t              st        j                  |      }|j	                  dd       }|j	                  dd       }| j
                  r$t        | j
                  j                  d            nd}g }g }| j
                  r| j
                  j                  dd      j                         }	t        |	      D ]T  \  }
}dt        |
      z   }|d|z   d|z   g}t        ||      }|j                  |       |j                  |       |P|||<   V dd	d
dddh|}t        |      j                  |      }|rt        d| d      t        |      |kD  rt        d      d ||h|vrt!        d      t        |t              r|j#                         n|}t        j$                  |      j'                         st)        d      |g|||S )Nr   r   ,r    r|  fix_r)   r*   r+   r,   zUnknown keyword arguments: r  zToo many positional arguments.r   r   )r9   r%   rJ   r   r7   shapesr  splitrO  	enumeratestrr   r  set
differencer.   r  r  r   r   r   )distr@   rA   r/   r   r   
num_shapesfshape_keysfshapesrZ  r  r  keynamesr7  
known_keysunknown_keys
uncensoreds                     r0   r  r    s   dL)zz$88FD!DXXh%F04T[[&&s+,JKG
 {{$$S#.446f% 	 DAqA,C#'6A:.E&tU3Cs#NN3S		  +x(2%02Jt9''
3L5l^1EFF
4y:899D&+7++  ' ( 	( &0l%C!J;;z"&&(?@@)7)D)&))r2   c                   R    e Zd ZdZej
                  Zd Zd Zd Z	d Z
d Zd Zd Zy	)
levy_genag  A Levy continuous random variable.

    %(before_notes)s

    See Also
    --------
    levy_stable, levy_l

    Notes
    -----
    The probability density function for `levy` is:

    .. math::

        f(x) = \frac{1}{\sqrt{2\pi x^3}} \exp\left(-\frac{1}{2x}\right)

    for :math:`x > 0`.

    This is the same as the Levy-stable distribution with :math:`a=1/2` and
    :math:`b=1`.

    %(after_notes)s

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.stats import levy
    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots(1, 1)

    Calculate the first four moments:

    >>> mean, var, skew, kurt = levy.stats(moments='mvsk')

    Display the probability density function (``pdf``):

    >>> # `levy` is very heavy-tailed.
    >>> # To show a nice plot, let's cut off the upper 40 percent.
    >>> a, b = levy.ppf(0), levy.ppf(0.6)
    >>> x = np.linspace(a, b, 100)
    >>> ax.plot(x, levy.pdf(x),
    ...        'r-', lw=5, alpha=0.6, label='levy pdf')

    Alternatively, the distribution object can be called (as a function)
    to fix the shape, location and scale parameters. This returns a "frozen"
    RV object holding the given parameters fixed.

    Freeze the distribution and display the frozen ``pdf``:

    >>> rv = levy()
    >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

    Check accuracy of ``cdf`` and ``ppf``:

    >>> vals = levy.ppf([0.001, 0.5, 0.999])
    >>> np.allclose([0.001, 0.5, 0.999], levy.cdf(vals))
    True

    Generate random numbers:

    >>> r = levy.rvs(size=1000)

    And compare the histogram:

    >>> # manual binning to ignore the tail
    >>> bins = np.concatenate((np.linspace(a, b, 20), [np.max(r)]))
    >>> ax.hist(r, bins=bins, density=True, histtype='stepfilled', alpha=0.2)
    >>> ax.set_xlim([x[0], x[-1]])
    >>> ax.legend(loc='best', frameon=False)
    >>> plt.show()

    c                     g S rH   r   rd   s    r0   re   zlevy_gen._shape_infoA  r   r2   c                     dt        j                  dt         j                  z  |z        z  |z  t        j                  dd|z  z        z  S Nr   rO   r  r  r   s     r0   rl   zlevy_gen._pdfD  s=    2771RUU719%%)BFF2qs8,<<<r2   c                 X    t        j                  t        j                  d|z              S r  )rq   erfcrJ   r   r   s     r0   ro   zlevy_gen._cdfH  s    wwrwwsQw'((r2   c                 X    t        j                  t        j                  d|z              S r  r	  r   s     r0   rs   zlevy_gen._sfL  s    vvbggcAg&''r2   c                 .    t        |dz        }d||z  z  S NrO   r   r   r?   rw   r7  s      r0   rx   zlevy_gen._ppfO  s    !ncCi  r2   c                 >    ddt        j                  |      dz  z  z  S r*  )rq   erfinvr  s     r0   r{   zlevy_gen._isfT  s    !BIIaL!O#$$r2   c                 ~    t         j                  t         j                  t         j                  t         j                  fS rH   r  rd   s    r0   r   zlevy_gen._statsW  r  r2   Nr}   r~   r   r   r   r  r  re   rl   ro   rs   rx   r{   r   r   r2   r0   rj  rj    s9    GP "44M=)(!
%.r2   rj  levyc                   R    e Zd ZdZej
                  Zd Zd Zd Z	d Z
d Zd Zd Zy	)

levy_l_gena  A left-skewed Levy continuous random variable.

    %(before_notes)s

    See Also
    --------
    levy, levy_stable

    Notes
    -----
    The probability density function for `levy_l` is:

    .. math::
        f(x) = \frac{1}{|x| \sqrt{2\pi |x|}} \exp{ \left(-\frac{1}{2|x|} \right)}

    for :math:`x < 0`.

    This is the same as the Levy-stable distribution with :math:`a=1/2` and
    :math:`b=-1`.

    %(after_notes)s

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.stats import levy_l
    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots(1, 1)

    Calculate the first four moments:

    >>> mean, var, skew, kurt = levy_l.stats(moments='mvsk')

    Display the probability density function (``pdf``):

    >>> # `levy_l` is very heavy-tailed.
    >>> # To show a nice plot, let's cut off the lower 40 percent.
    >>> a, b = levy_l.ppf(0.4), levy_l.ppf(1)
    >>> x = np.linspace(a, b, 100)
    >>> ax.plot(x, levy_l.pdf(x),
    ...        'r-', lw=5, alpha=0.6, label='levy_l pdf')

    Alternatively, the distribution object can be called (as a function)
    to fix the shape, location and scale parameters. This returns a "frozen"
    RV object holding the given parameters fixed.

    Freeze the distribution and display the frozen ``pdf``:

    >>> rv = levy_l()
    >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

    Check accuracy of ``cdf`` and ``ppf``:

    >>> vals = levy_l.ppf([0.001, 0.5, 0.999])
    >>> np.allclose([0.001, 0.5, 0.999], levy_l.cdf(vals))
    True

    Generate random numbers:

    >>> r = levy_l.rvs(size=1000)

    And compare the histogram:

    >>> # manual binning to ignore the tail
    >>> bins = np.concatenate(([np.min(r)], np.linspace(a, b, 20)))
    >>> ax.hist(r, bins=bins, density=True, histtype='stepfilled', alpha=0.2)
    >>> ax.set_xlim([x[0], x[-1]])
    >>> ax.legend(loc='best', frameon=False)
    >>> plt.show()

    c                     g S rH   r   rd   s    r0   re   zlevy_l_gen._shape_info  r   r2   c                     t        |      }dt        j                  dt        j                  z  |z        z  |z  t        j                  dd|z  z        z  S rm  )r  rJ   r   r   r   r?   rk   r  s      r0   rl   zlevy_l_gen._pdf  sF    V255$$R'r1R4y(999r2   c                 f    t        |      }dt        dt        j                  |      z        z  dz
  S r  )r  r   rJ   r   r}  s      r0   ro   zlevy_l_gen._cdf  s,    V9Q_--11r2   c                 `    t        |      }dt        dt        j                  |      z        z  S r  )r  r   rJ   r   r}  s      r0   rs   zlevy_l_gen._sf  s'    V8AO,,,r2   c                 4    t        |dz   dz        }d||z  z  S )Nr   rO   r3  r   rs  s      r0   rx   zlevy_l_gen._ppf  s#    SA&sSy!!r2   c                 *    dt        |dz        dz  z  S )Nr  rO   r   r  s     r0   r{   zlevy_l_gen._isf  s    )AaC.!###r2   c                 ~    t         j                  t         j                  t         j                  t         j                  fS rH   r  rd   s    r0   r   zlevy_l_gen._stats  r  r2   Nrw  r   r2   r0   rz  rz  ^  s9    FN "44M:
2-"$.r2   rz  levy_lc                        e Zd ZdZd ZddZd Zd Zd Zd Z	d Z
d	 Zd
 Zd Zd Zd Ze ee       fd              Z xZS )logistic_gena  A logistic (or Sech-squared) continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `logistic` is:

    .. math::

        f(x) = \frac{\exp(-x)}
                    {(1+\exp(-x))^2}

    `logistic` is a special case of `genlogistic` with ``c=1``.

    Remark that the survival function (``logistic.sf``) is equal to the
    Fermi-Dirac distribution describing fermionic statistics.

    %(after_notes)s

    %(example)s

    c                     g S rH   r   rd   s    r0   re   zlogistic_gen._shape_info  r   r2   c                 &    |j                  |      S r  )logisticr   s      r0   r   zlogistic_gen._rvs  s    $$$$//r2   c                 J    t        j                  | j                  |            S rH   r  r   s     r0   rl   zlogistic_gen._pdf  r  r2   c                     t        j                  |       }|dt        j                  t        j                  |            z  z
  S r7  )rJ   r  rq   r  r   )r?   rk   r^  s      r0   r   zlogistic_gen._logpdf  s2    VVAYJ2++++r2   c                 ,    t        j                  |      S rH   r  r   s     r0   ro   zlogistic_gen._cdf      xx{r2   c                 ,    t        j                  |      S rH   rq   	log_expitr   s     r0   r   zlogistic_gen._logcdf  s    ||Ar2   c                 ,    t        j                  |      S rH   r  r   s     r0   rx   zlogistic_gen._ppf  r  r2   c                 .    t        j                  |       S rH   r  r   s     r0   rs   zlogistic_gen._sf  s    xx|r2   c                 .    t        j                  |       S rH   r  r   s     r0   r   zlogistic_gen._logsf  s    ||QBr2   c                 .    t        j                  |       S rH   r  r   s     r0   r{   zlogistic_gen._isf  s    |r2   c                 R    dt         j                  t         j                  z  dz  ddfS )Nr   r;  g333333?r(  rd   s    r0   r   zlogistic_gen._stats  s!    "%%+c/1g--r2   c                      yr7  r   rd   s    r0   r   zlogistic_gen._entropy  s    r2   c                   
 |j                  dd      rt        |   g|i |S t        | ||      \  }}t	              | j                        \  }}|j                  d|      |j                  d|      }}|ffd	
|ffd	
fd}|+|)t        j                  
|f      }	|	j                  d   }|}nT|+|)t        j                  |f      }	|	j                  d   }|}n't        j                  |||f      }	|	j                  \  }}t        |      }|	j                  r||fS t        |   g|i |S )	Nr  Fr)   r*   c                 p    | z
  |z  }t        j                  t        j                  |            dz  z
  S r  )rJ   r  rq   r  )r)   r*   r  r@   r\   s      r0   dl_dlocz!logistic_gen.fit.<locals>.dl_dloc  s1    u$A66"((1+&1,,r2   c                 v    |z
  | z  }t        j                  |t        j                  |dz        z        z
  S r  )rJ   r  r  )r*   r)   r  r@   r\   s      r0   	dl_dscalez#logistic_gen.fit.<locals>.dl_dscale  s5    u$A66!BGGAaCL.)A--r2   c                 2    | \  }} ||       ||      fS rH   r   )paramsr)   r*   r  r  s      r0   rW  zlogistic_gen.fit.<locals>.func!  s%    JC3&	%(===r2   r   )r-   r;   r=   r  r  r  r7   r   r  rk   r  success)r?   r@   rA   r/   r   r   r)   r*   rW  r  r  r  r\   r  s    `        @@@r0   r=   zlogistic_gen.fit  sS    88J&7;t3d3d338t9=tEdFI ^^D)
UXXeS)488GU+CU  & 	- "& 	.	> $,--#0C%%(CE&.--	E84CEE!HEC--sEl3CJC E
 #e 	7W[555	7r2   r   )r}   r~   r   r   re   r   rl   r   ro   r   rx   rs   r   r{   r   r   rE   r
   r   r=   r  r  s   @r0   r  r    se    .0', . M*07 + 07r2   r  r  c                   N    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 Zd Zd Zy)loggamma_gena  A log gamma continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `loggamma` is:

    .. math::

        f(x, c) = \frac{\exp(c x - \exp(x))}
                       {\Gamma(c)}

    for all :math:`x, c > 0`. Here, :math:`\Gamma` is the
    gamma function (`scipy.special.gamma`).

    `loggamma` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zloggamma_gen._shape_infoV  r  r2   Nc                     t        j                  |j                  |dz   |            t        j                  |j                  |            |z  z   S )Nr   r  )rJ   r   r  r  r  s       r0   r   zloggamma_gen._rvsY  sM     |))!a%d);<&&--4-89!;< 	=r2   c                     t        j                  ||z  t        j                  |      z
  t        j                  |      z
        S rH   rJ   r   rq   r  r  s      r0   rl   zloggamma_gen._pdfg  s.    vvac"&&)mBJJqM122r2   c                 d    ||z  t        j                  |      z
  t        j                  |      z
  S rH   r  r  s      r0   r   zloggamma_gen._logpdfk  s%    sRVVAYA..r2   c                 6    t        |t        k  ||fd d       S )Nc                 d    t        j                  || z  t        j                  |dz         z
        S rX   r  rc  s     r0   r  z#loggamma_gen._cdf.<locals>.<lambda>~  s$    rvvacBJJqsO.C'D r2   c                 T    t        j                  |t        j                  |             S rH   )rq   r  rJ   r   rc  s     r0   r  z#loggamma_gen._cdf.<locals>.<lambda>  s    "++a*C r2   r  r   r   r  s      r0   ro   zloggamma_gen._cdfn  s%     !h,ADCE 	Er2   c                 d    t        j                  ||      }t        |t        k  |||fd d       S )Nc                 d    t        j                  |      t        j                  |dz         z   |z  S rX   r3  r  rw   r  s      r0   r  z#loggamma_gen._ppf.<locals>.<lambda>  s$    266!9rzz!A#+F*I r2   c                 ,    t        j                  |       S rH   r+  r  s      r0   r  z#loggamma_gen._ppf.<locals>.<lambda>      RVVAY r2   r  )rq   r  r   r   r?   rw   r  r  s       r0   rx   zloggamma_gen._ppf  s5     NN1a !e)aAYI68 	8r2   c                 6    t        |t        k  ||fd d       S )Nc                 f    t        j                  || z  t        j                  |dz         z
         S rX   )rJ   r  rq   r  rc  s     r0   r  z"loggamma_gen._sf.<locals>.<lambda>  s'    1rzz!A#1F(G'G r2   c                 T    t        j                  |t        j                  |             S rH   )rq   r  rJ   r   rc  s     r0   r  z"loggamma_gen._sf.<locals>.<lambda>  s    ",,q"&&)*D r2   r  r  r  s      r0   rs   zloggamma_gen._sf  s#    !h,AGDF 	Fr2   c                 d    t        j                  ||      }t        |t        k  |||fd d       S )Nc                 f    t        j                  |       t        j                  |dz         z   |z  S rX   )rJ   r  rq   r  r  s      r0   r  z#loggamma_gen._isf.<locals>.<lambda>  s&    288QB<"**QqS/+I1*L r2   c                 ,    t        j                  |       S rH   r+  r  s      r0   r  z#loggamma_gen._isf.<locals>.<lambda>  r  r2   r  )rq   r  r   r   r  s       r0   r{   zloggamma_gen._isf  s5     OOAq!!e)aAYL68 	8r2   c                     t        j                  |      }t        j                  d|      }t        j                  d|      t        j                  |d      z  }t        j                  d|      ||z  z  }||||fS )Nr   rO   r  r  )rq   r  	polygammarJ   r  )r?   r  r   r  r  excess_kurtosiss         r0   r   zloggamma_gen._stats  si     zz!}ll1a <<1%c(::,,q!,C8S(O33r2   c                 8    d }d }t        |dk\  |f||      }|S )Nc                 h    t        j                  |       | t        j                  |       z  z
  | z   }|S rH   )rq   r  r  )r  r  s     r0   r  z&loggamma_gen._entropy.<locals>.regular  s+    

1BJJqM 11A5AHr2   c                     dt        j                  |       z  | dz  dz  z   | dz  dz  z
  | dz  dz  z   }t        j                         |z   }|S )Nr  r3  r  r  r  r     )rJ   r   r   r   )r  termr  s      r0   r$  z)loggamma_gen._entropy.<locals>.asymptotic  sO    q	>AsF1H,q#vby81c6#:ED$&AHr2   -   r&  r  )r?   r  r  r$  r  s        r0   r   zloggamma_gen._entropy  s)    		 qBw@r2   r   r}   r~   r   r   re   r   rl   r   ro   rx   rs   r{   r   r   r   r2   r0   r  r  =  s<    0E=3/E&8F84r2   r  loggammac                   r     e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Ze ee       fd
              Z xZS )loglaplace_genaT  A log-Laplace continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `loglaplace` is:

    .. math::

        f(x, c) = \begin{cases}\frac{c}{2} x^{ c-1}  &\text{for } 0 < x < 1\\
                               \frac{c}{2} x^{-c-1}  &\text{for } x \ge 1
                  \end{cases}

    for :math:`c > 0`.

    `loglaplace` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    Suppose a random variable ``X`` follows the Laplace distribution with
    location ``a`` and scale ``b``.  Then ``Y = exp(X)`` follows the
    log-Laplace distribution with ``c = 1 / b`` and ``scale = exp(a)``.

    References
    ----------
    T.J. Kozubowski and K. Podgorski, "A log-Laplace growth rate model",
    The Mathematical Scientist, vol. 28, pp. 49-60, 2003.

    %(example)s

    c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zloglaplace_gen._shape_info  r  r2   c                 X    |dz  }t        j                  |dk  ||       }|||dz
  z  z  S r  rJ   rF  )r?   rk   r  cd2s       r0   rl   zloglaplace_gen._pdf  s7     eHHQUAr"1qs8|r2   c                 V    t        j                  |dk  d||z  z  dd|| z  z  z
        S Nr   r   r  r  s      r0   ro   zloglaplace_gen._cdf  s/    xxAs1a4x3qA2w;77r2   c                 V    t        j                  |dk  dd||z  z  z
  d|| z  z        S r  r  r  s      r0   rs   zloglaplace_gen._sf  s/    xxAq3q!t8|SaR[99r2   c                 `    t        j                  |dk  d|z  d|z  z  dd|z
  z  d|z  z        S Nr   r   r   rO   r3  r  r  s      r0   rx   zloglaplace_gen._ppf  s7    xxC#a%3q5!1As1uIa3HIIr2   c                 `    t        j                  |dkD  dd|z
  z  d|z  z  d|z  d|z  z        S r  r  r  s      r0   r{   zloglaplace_gen._isf  s6    xxC#sQw-3q5!9AaC46?KKr2   c                     t        j                  d      5  |dz  |dz  }}t        j                  ||k  |||z
  z  t         j                        cd d d        S # 1 sw Y   y xY w)Nr2  r3  rO   )rJ   r5  rF  rc   )r?   r\   r  r  n2s        r0   r   zloglaplace_gen._munp  sT    [[) 	=T1a4B88BGR27^RVV<	= 	= 	=s   8AA"c                 8    t        j                  d|z        dz   S r  r+  rr  s     r0   r   zloglaplace_gen._entropy  s    vvc!e}s""r2   c                    t        | |||      \  }}}}|t        t        |       |   |g|i |S t	        j
                  ||k        rt        d|t        j                        |dk7  r||z
  }t        j                  t	        j                  |      |t	        j                  |      nd |d|z  nd d      \  }}|}	|t	        j                  |      n|}
|d|z  n|}||	|
fS )N
loglaplacer  r   r   r5   )r   r   r,   )r  r;   r<   r=   rJ   r  rD  rc   r5  r   r   )r?   r@   rA   r/   r  r   r   r   r   r)   r*   r  r  s               r0   r=   zloglaplace_gen.fit  s     "=T4=A4"Ib$ <dT.tCdCdCC 66$$,|4rvvFF 19$;D {{266$<282Dv$*,.!B$d"'  )1 #^q	ZAER#u}r2   )r}   r~   r   r   re   rl   ro   rs   rx   r{   r   r   rE   r
   r   r=   r  r  s   @r0   r  r    sU    @E8:JL=
# M* + r2   r  r  c                 H    t        | dk7  | |fd t        j                         S )Nr   c                     t        j                  |       dz   d|dz  z  z  t        j                  || z  t        j                  dt         j                  z        z        z
  S r  )rJ   r   r   r   rk   r  s     r0   r  z!_lognorm_logpdf.<locals>.<lambda>  sM    RVVAY\MQAX$>&(ffQURWWQY5G-G&H%I r2   r  r  s     r0   _lognorm_logpdfr    s*    a1fq!fJvvg r2   c                        e Zd ZdZej
                  Zd ZddZd Z	d Z
d Zd Zd Zd	 Zd
 Zd Zd Zd Ze eed       fd              Z xZS )lognorm_gena  A lognormal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `lognorm` is:

    .. math::

        f(x, s) = \frac{1}{s x \sqrt{2\pi}}
                  \exp\left(-\frac{\log^2(x)}{2s^2}\right)

    for :math:`x > 0`, :math:`s > 0`.

    `lognorm` takes ``s`` as a shape parameter for :math:`s`.

    %(after_notes)s

    Suppose a normally distributed random variable ``X`` has  mean ``mu`` and
    standard deviation ``sigma``. Then ``Y = exp(X)`` is lognormally
    distributed with ``s = sigma`` and ``scale = exp(mu)``.

    %(example)s

    The logarithm of a log-normally distributed random variable is
    normally distributed:

    >>> import numpy as np
    >>> import matplotlib.pyplot as plt
    >>> from scipy import stats
    >>> fig, ax = plt.subplots(1, 1)
    >>> mu, sigma = 2, 0.5
    >>> X = stats.norm(loc=mu, scale=sigma)
    >>> Y = stats.lognorm(s=sigma, scale=np.exp(mu))
    >>> x = np.linspace(*X.interval(0.999))
    >>> y = Y.rvs(size=10000)
    >>> ax.plot(x, X.pdf(x), label='X (pdf)')
    >>> ax.hist(np.log(y), density=True, bins=x, label='log(Y) (histogram)')
    >>> ax.legend()
    >>> plt.show()

    c                 @    t        dddt        j                  fd      gS )Nr  Fr   r  rb   rd   s    r0   re   zlognorm_gen._shape_infoJ  r  r2   c                 P    t        j                  ||j                  |      z        S rH   rJ   r   r   )r?   r  r   r   s       r0   r   zlognorm_gen._rvsM  s!    vva,66t<<==r2   c                 L    t        j                  | j                  ||            S rH   r  r?   rk   r  s      r0   rl   zlognorm_gen._pdfP  rx  r2   c                     t        ||      S rH   r  r  s      r0   r   zlognorm_gen._logpdfT  s    q!$$r2   c                 D    t        t        j                  |      |z        S rH   r   rJ   r   r  s      r0   ro   zlognorm_gen._cdfW  s    Q''r2   c                 D    t        t        j                  |      |z        S rH   rN  r  s      r0   r   zlognorm_gen._logcdfZ  s    BFF1IM**r2   c                 D    t        j                  |t        |      z        S rH   rJ   r   r   r?   rw   r  s      r0   rx   zlognorm_gen._ppf]      vva)A,&''r2   c                 D    t        t        j                  |      |z        S rH   r   rJ   r   r  s      r0   rs   zlognorm_gen._sf`  s    q	A&&r2   c                 D    t        t        j                  |      |z        S rH   )r   rJ   r   r  s      r0   r   zlognorm_gen._logsfc  s    266!9q=))r2   c                 D    t        j                  |t        |      z        S rH   rJ   r   r   r  s      r0   r{   zlognorm_gen._isff  r  r2   c                     t        j                  ||z        }t        j                  |      }||dz
  z  }t        j                  |dz
        d|z   z  }t        j                  g d|      }||||fS Nr   rO   )r   rO   r  r   r  )rJ   r   r   polyval)r?   r  r  r<  r=  r>  r?  s          r0   r   zlognorm_gen._statsi  sf    FF1Q3KWWQZ1gWWQqS\1Q3ZZ*A.3Br2   c                     ddt        j                  dt         j                  z        z   dt        j                  |      z  z   z  S Nr   r   rO   r   )r?   r  s     r0   r   zlognorm_gen._entropyq  s3    a"&&255/)Aq	M9::r2   aF          When `method='MLE'` and
        the location parameter is fixed by using the `floc` argument,
        this function uses explicit formulas for the maximum likelihood
        estimation of the log-normal shape and scale parameters, so the
        `optimizer`, `loc` and `scale` keyword arguments are ignored.
        If the location is free, a likelihood maximum is found by
        setting its partial derivative wrt to location to 0, and
        solving by substituting the analytical expressions of shape
        and scale (or provided parameters).
        See, e.g., equation 3.1 in
        A. Clifford Cohen & Betty Jones Whitten (1980)
        Estimation in the Three-Parameter Lognormal Distribution,
        Journal of the American Statistical Association, 75:370, 399-404
        https://doi.org/10.2307/2287466
        

r   c                     |j                  dd      rt           g|i |S t         ||      }|\  }t	        j
                        }fdfd} fd}|&t	        j                  |      }	||	z
  }
 ||
      } ||
      }d|	z  }|dk\  r||z
  }
 ||
      }|dz  }|dk\  rt	        j                  |
      rt	        j                  |      st           g|i |S t	        j                  t	        j                  |
t        j                         |
dz
        } ||      }d|
|z
  z  }t	        j                  |      rt	        j                  |      rt	        j                  |      t	        j                  |      k(  rh|
|z
  } ||      }|dz  }t	        j                  |      rAt	        j                  |      r,t	        j                  |      t	        j                  |      k(  rht	        j                  |      rt	        j                  |      st           g|i |S t        |||
f	      }|j                  st           g|i |S  ||j                        }||kD  r|j                  n||	z
  }n#||k\  rt        d
dt        j                        |} |      \  }} j!                  |      r|dkD  st           g|i |S |||fS )Nr  Fc                    t        j                  | z
        }xs# t        j                  j                               }xsA t        j                  t        j                  t        j                  |      z
  dz              }||fS r  )rJ   r   r   r   r   )r)   lndatar*   rn  r@   r   fshapes       r0   get_shape_scalez(lognorm_gen.fit.<locals>.get_shape_scale  sq     ~s
+3bffV[[]3EKbggbggvu/E.I&JKE%<r2   c                      |       \  }}| z
  }t        j                  dt        j                  ||z        |dz  z  z   |z        S r*  rJ   r  r   )r)   rn  r*   shiftedr@   r  s       r0   dL_dLocz lognorm_gen.fit.<locals>.dL_dLoc  sI    *3/LE5SjG661rvvgem4UAX==wFGGr2   c                 F     |       \  }}j                  || |f       S rH   )nnlf)r)   rn  r*   r@   r  r?   s      r0   llzlognorm_gen.fit.<locals>.ll  s,    *3/LE5IIuc514888r2   rO   gưr   r  lognormr   r  r   )r-   r;   r=   r  rJ   r@  spacingr   r  	nextafterrc   rK   r&   	convergedr  rD  r]   )r?   r@   rA   r/   
parametersr   rB  r  r  r  rM   dL_dLoc_rbrack	ll_rbrackr  rL   dL_dLoc_lbrackr  ll_rootr)   rn  r*   r   r  r  r  s   ``                   @@@r0   r=   zlognorm_gen.fitt  s   $ 88J&7;t3d3d330tT4H
%/"fdF66$<	 	H	9
 < jj*G'F %V_N6
IKE E)!E)!(
 !E)
 ;;v&bkk..I w{47$7$77
 ZZVbffW =vaxHF$V_N&)E;;v&2;;~+Fww~."''.2II%!(
	 ;;v&2;;~+Fww~."''.2II ;;v&bkk..Iw{47$7$77 g/?@C==w{47$7$77
 lG%	1#((x7GC x"9BbffEEC&s+uu%%!)7;t3d3d33c5  r2   r   )r}   r~   r   r   r   r  r  re   r   rl   r   ro   r   rx   rs   r   r{   r   r   rE   r   r=   r  r  s   @r0   r  r    s}    *V "44ME>*%(+('*(; } 5  Z!! "Z!r2   r  r  c                   f    e Zd ZdZej
                  Zd ZddZd Z	d Z
d Zd Zd	 Zd
 Zd Zd Zy)
gibrat_genaB  A Gibrat continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `gibrat` is:

    .. math::

        f(x) = \frac{1}{x \sqrt{2\pi}} \exp(-\frac{1}{2} (\log(x))^2)

    `gibrat` is a special case of `lognorm` with ``s=1``.

    %(after_notes)s

    %(example)s

    c                     g S rH   r   rd   s    r0   re   zgibrat_gen._shape_info  r   r2   Nc                 J    t        j                  |j                  |            S rH   r  r   s      r0   r   zgibrat_gen._rvs  s    vvl224899r2   c                 J    t        j                  | j                  |            S rH   r  r   s     r0   rl   zgibrat_gen._pdf  r  r2   c                     t        |d      S r  r  r   s     r0   r   zgibrat_gen._logpdf  s    q#&&r2   c                 >    t        t        j                  |            S rH   r  r   s     r0   ro   zgibrat_gen._cdf  s    ##r2   c                 >    t        j                  t        |            S rH   r  r   s     r0   rx   zgibrat_gen._ppf      vvil##r2   c                 >    t        t        j                  |            S rH   r  r   s     r0   rs   zgibrat_gen._sf  s    q	""r2   c                 >    t        j                  t        |            S rH   r  r  s     r0   r{   zgibrat_gen._isf  r  r2   c                     t         j                  }t        j                  |      }||dz
  z  }t        j                  |dz
        d|z   z  }t        j                  g d|      }||||fS r  )rJ   er   r  )r?   r  r<  r=  r>  r?  s         r0   r   zgibrat_gen._stats  s^    DDWWQZ1q5kWWQU^q1u%ZZ*A.3Br2   c                 Z    dt        j                  dt         j                  z        z  dz   S r  r   rd   s    r0   r   zgibrat_gen._entropy  s#    RVVAI&&,,r2   r   )r}   r~   r   r   r   r  r  re   r   rl   r   ro   rx   rs   r{   r   r   r   r2   r0   r  r    sF    & "44M:''$$#$-r2   r  gibratc                   N    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 Zd Zd Zy)maxwell_gena  A Maxwell continuous random variable.

    %(before_notes)s

    Notes
    -----
    A special case of a `chi` distribution,  with ``df=3``, ``loc=0.0``,
    and given ``scale = a``, where ``a`` is the parameter used in the
    Mathworld description [1]_.

    The probability density function for `maxwell` is:

    .. math::

        f(x) = \sqrt{2/\pi}x^2 \exp(-x^2/2)

    for :math:`x >= 0`.

    %(after_notes)s

    References
    ----------
    .. [1] http://mathworld.wolfram.com/MaxwellDistribution.html

    %(example)s
    c                     g S rH   r   rd   s    r0   re   zmaxwell_gen._shape_info>  r   r2   Nc                 2    t         j                  d||      S )Nr;  r  r  r  r   s      r0   r   zmaxwell_gen._rvsA  s    wwsLwAAr2   c                 T    t         |z  |z  t        j                  | |z  dz        z  S r7  )r#   rJ   r   r   s     r0   rl   zmaxwell_gen._pdfD  s*    q "2661"Q$s(#333r2   c                     t        j                  d      5  t        dt        j                  |      z  z   d|z  |z  z
  cd d d        S # 1 sw Y   y xY w)Nr2  r3  rO   r   )rJ   r5  r$   r   r   s     r0   r   zmaxwell_gen._logpdfH  sD    [[) 	?&266!94s1uQw>	? 	? 	?s   (A		Ac                 :    t        j                  d||z  dz        S Nr  r   r  r   s     r0   ro   zmaxwell_gen._cdfM  s    {{3!C((r2   c                 Z    t        j                  dt        j                  d|      z        S r  r  r   s     r0   rx   zmaxwell_gen._ppfP  s!    wwqQ//00r2   c                 :    t        j                  d||z  dz        S r  r  r   s     r0   rs   zmaxwell_gen._sfS  s    ||C1S))r2   c                 Z    t        j                  dt        j                  d|      z        S r  r  r   s     r0   r{   zmaxwell_gen._isfV  s!    wwqa0011r2   c                    dt         j                  z  dz
  }dt        j                  dt         j                  z        z  ddt         j                  z  z
  t        j                  d      ddt         j                  z  z
  z  |dz  z  dt         j                  z  t         j                  z  d	t         j                  z  z   d
z
  |dz  z  fS )Nr  r'  rO   r       r  r  r     i  rJ   r   r   r?   r7  s     r0   r   zmaxwell_gen._statsY  s    gai"''#bee)$$!BEE'	
Br"%%xK(c1RUU2553ruu9,s2c3h>@ 	@r2   c                 h    t         dt        j                  dt        j                  z        z  z   dz
  S r  )r    rJ   r   r   rd   s    r0   r   zmaxwell_gen._entropy`  s'    BFF1RUU7O++C//r2   r   r  r   r2   r0   r  r  #  s;    4B4?
)1*2@0r2   r  maxwellc                   4    e Zd ZdZd Zd Zd Zd Zd Zd Z	y)	
mielke_gena  A Mielke Beta-Kappa / Dagum continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `mielke` is:

    .. math::

        f(x, k, s) = \frac{k x^{k-1}}{(1+x^s)^{1+k/s}}

    for :math:`x > 0` and :math:`k, s > 0`. The distribution is sometimes
    called Dagum distribution ([2]_). It was already defined in [3]_, called
    a Burr Type III distribution (`burr` with parameters ``c=s`` and
    ``d=k/s``).

    `mielke` takes ``k`` and ``s`` as shape parameters.

    %(after_notes)s

    References
    ----------
    .. [1] Mielke, P.W., 1973 "Another Family of Distributions for Describing
           and Analyzing Precipitation Data." J. Appl. Meteor., 12, 275-280
    .. [2] Dagum, C., 1977 "A new model for personal income distribution."
           Economie Appliquee, 33, 327-367.
    .. [3] Burr, I. W. "Cumulative frequency functions", Annals of
           Mathematical Statistics, 13(2), pp 215-232 (1942).

    %(example)s

    c                     t        dddt        j                  fd      }t        dddt        j                  fd      }||gS )Nr
  Fr   r  r  rb   )r?   iki_ss      r0   re   zmielke_gen._shape_info  <    UQK@ea[.ACyr2   c                 B    |||dz
  z  z  d||z  z   d|dz  |z  z   z  z  S r  r   r?   rk   r
  r  s       r0   rl   zmielke_gen._pdf  s2    QsU|s1a4x3quQw;777r2   c                     t        j                  d      5  t        j                  |      t        j                  |      |dz
  z  z   t        j                  ||z        d||z  z   z  z
  cd d d        S # 1 sw Y   y xY w)Nr2  r3  r   )rJ   r5  r   r  r-  s       r0   r   zmielke_gen._logpdf  sf    [[) 	L66!9rvvay!a%00288AqD>1qs73KK	L 	L 	Ls   AA44A=c                 0    ||z  d||z  z   |dz  |z  z  z  S r  r   r-  s       r0   ro   zmielke_gen._cdf  s&    !ts1a4x1S57+++r2   c                 P    t        ||dz  |z        }t        |d|z
  z  d|z        S r  r  )r?   rw   r
  r  qsks        r0   rx   zmielke_gen._ppf  s.    !QsU1Wo3C=#a%((r2   c                 L    d }t        ||k  |||f|t        j                        S )Nc                     t        j                  || z   |z        t        j                  d| |z  z
        z  t        j                  ||z        z  S rX   r"  )r\   r
  r  s      r0   r  z$mielke_gen._munp.<locals>.nth_moment  s?    88QqS!G$RXXa!e_4RXXac]BBr2   r  )r?   r\   r
  r  r  s        r0   r   zmielke_gen._munp  s)    	C !a%!QJ??r2   N)
r}   r~   r   r   re   rl   r   ro   rx   r   r   r2   r0   r'  r'  g  s(     B
8L
,)@r2   r'  mielkec                   R    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zd Zd Zy)
kappa4_genap  Kappa 4 parameter distribution.

    %(before_notes)s

    Notes
    -----
    The probability density function for kappa4 is:

    .. math::

        f(x, h, k) = (1 - k x)^{1/k - 1} (1 - h (1 - k x)^{1/k})^{1/h-1}

    if :math:`h` and :math:`k` are not equal to 0.

    If :math:`h` or :math:`k` are zero then the pdf can be simplified:

    h = 0 and k != 0::

        kappa4.pdf(x, h, k) = (1.0 - k*x)**(1.0/k - 1.0)*
                              exp(-(1.0 - k*x)**(1.0/k))

    h != 0 and k = 0::

        kappa4.pdf(x, h, k) = exp(-x)*(1.0 - h*exp(-x))**(1.0/h - 1.0)

    h = 0 and k = 0::

        kappa4.pdf(x, h, k) = exp(-x)*exp(-exp(-x))

    kappa4 takes :math:`h` and :math:`k` as shape parameters.

    The kappa4 distribution returns other distributions when certain
    :math:`h` and :math:`k` values are used.

    +------+-------------+----------------+------------------+
    | h    | k=0.0       | k=1.0          | -inf<=k<=inf     |
    +======+=============+================+==================+
    | -1.0 | Logistic    |                | Generalized      |
    |      |             |                | Logistic(1)      |
    |      |             |                |                  |
    |      | logistic(x) |                |                  |
    +------+-------------+----------------+------------------+
    |  0.0 | Gumbel      | Reverse        | Generalized      |
    |      |             | Exponential(2) | Extreme Value    |
    |      |             |                |                  |
    |      | gumbel_r(x) |                | genextreme(x, k) |
    +------+-------------+----------------+------------------+
    |  1.0 | Exponential | Uniform        | Generalized      |
    |      |             |                | Pareto           |
    |      |             |                |                  |
    |      | expon(x)    | uniform(x)     | genpareto(x, -k) |
    +------+-------------+----------------+------------------+

    (1) There are at least five generalized logistic distributions.
        Four are described here:
        https://en.wikipedia.org/wiki/Generalized_logistic_distribution
        The "fifth" one is the one kappa4 should match which currently
        isn't implemented in scipy:
        https://en.wikipedia.org/wiki/Talk:Generalized_logistic_distribution
        https://www.mathwave.com/help/easyfit/html/analyses/distributions/gen_logistic.html
    (2) This distribution is currently not in scipy.

    References
    ----------
    J.C. Finney, "Optimization of a Skewed Logistic Distribution With Respect
    to the Kolmogorov-Smirnov Test", A Dissertation Submitted to the Graduate
    Faculty of the Louisiana State University and Agricultural and Mechanical
    College, (August, 2004),
    https://digitalcommons.lsu.edu/gradschool_dissertations/3672

    J.R.M. Hosking, "The four-parameter kappa distribution". IBM J. Res.
    Develop. 38 (3), 25 1-258 (1994).

    B. Kumphon, A. Kaew-Man, P. Seenoi, "A Rainfall Distribution for the Lampao
    Site in the Chi River Basin, Thailand", Journal of Water Resource and
    Protection, vol. 4, 866-869, (2012).
    :doi:`10.4236/jwarp.2012.410101`

    C. Winchester, "On Estimation of the Four-Parameter Kappa Distribution", A
    Thesis Submitted to Dalhousie University, Halifax, Nova Scotia, (March
    2000).
    http://www.nlc-bnc.ca/obj/s4/f2/dsk2/ftp01/MQ57336.pdf

    %(after_notes)s

    %(example)s

    c                 v    t        j                  ||      d   j                  }t        j                  |d      S )Nr   T
fill_value)rJ   r  rn  full)r?   r  r
  rn  s       r0   r]   zkappa4_gen._argcheck  s0    ##Aq)!,22wwu..r2   c                     t        ddt        j                   t        j                  fd      }t        ddt        j                   t        j                  fd      }||gS )Nr  Fr  r
  rb   )r?   ihr)  s      r0   re   zkappa4_gen._shape_info  sI    UbffWbff$5~FUbffWbff$5~FBxr2   c           
      
   t        j                  |dkD  |dkD        t        j                  |dkD  |dk(        t        j                  |dkD  |dk        t        j                  |dk  |dkD        t        j                  |dk  |dk(        t        j                  |dk  |dk        g}d }d }d }d }t        |||||||g||gt         j                        }d }d }t        |||||||g||gt         j                        }	||	fS )	Nr   c                 <    dt        j                  | |       z
  |z  S r  )rJ   rd  r  r
  s     r0   r  z#kappa4_gen._get_support.<locals>.f0  s    "..QB//22r2   c                 ,    t        j                  |       S rH   r+  r?  s     r0   r  z#kappa4_gen._get_support.<locals>.f1  s    66!9r2   c                 ~    t        j                  t        j                  |             }t         j                   |d d  |S rH   rJ   r  rn  rc   r  r
  r   s      r0   f3z#kappa4_gen._get_support.<locals>.f3  s,    !%AFF7AaDHr2   c                     d|z  S r  r   r?  s     r0   f5z#kappa4_gen._get_support.<locals>.f5      q5Lr2   defaultc                     d|z  S r  r   r?  s     r0   r  z#kappa4_gen._get_support.<locals>.f0%  rG  r2   c                 |    t        j                  t        j                  |             }t         j                  |d d  |S rH   rB  rC  s      r0   r  z#kappa4_gen._get_support.<locals>.f1(  s*    !%A66AaDHr2   rJ   r8  r   r  )
r?   r  r
  condlistr  r  rD  rF  r  r  s
             r0   r   zkappa4_gen._get_support
  s   NN1q5!a%0NN1q5!q&1NN1q5!a%0NN161q51NN16162NN161q513	3		
	 b"b"b1Q!#)
		
 b"b"b1Q!#) 2vr2   c                 N    t        j                  | j                  |||            S rH   r  r?   rk   r  r
  s       r0   rl   zkappa4_gen._pdf3  rg  r2   c                 >   t        j                  |dk7  |dk7        t        j                  |dk(  |dk7        t        j                  |dk7  |dk(        t        j                  |dk(  |dk(        g}d }d }d }d }t        |||||g|||gt         j                        S )Nr   c                     t        j                  d|z  dz
  | | z        t        j                  d|z  dz
  | d|| z  z
  d|z  z  z        z   S )zpdf = (1.0 - k*x)**(1.0/k - 1.0)*(
                      1.0 - h*(1.0 - k*x)**(1.0/k))**(1.0/h-1.0)
               logpdf = ...
            r   r]  rk   r  r
  s      r0   r  zkappa4_gen._logpdf.<locals>.f0>  sZ    
 JJs1us{QBqD1JJs1us{QBac	SU/C,CDE Fr2   c                 `    t        j                  d|z  dz
  | | z        d|| z  z
  d|z  z  z
  S )z~pdf = (1.0 - k*x)**(1.0/k - 1.0)*np.exp(-(
                      1.0 - k*x)**(1.0/k))
               logpdf = ...
            r   r]  rR  s      r0   r  zkappa4_gen._logpdf.<locals>.f1F  s9    
 ::c!eckA2a40C!A#IQ3GGGr2   c                 r    |  t        j                  d|z  dz
  | t        j                  |        z        z   S )z]pdf = np.exp(-x)*(1.0 - h*np.exp(-x))**(1.0/h - 1.0)
               logpdf = ...
            r   )rq   rq  rJ   r   rR  s      r0   r  zkappa4_gen._logpdf.<locals>.f2M  s4     2

3q53;2661":>>>r2   c                 6    |  t        j                  |        z
  S )zDpdf = np.exp(-x-np.exp(-x))
               logpdf = ...
            r.  rR  s      r0   rD  zkappa4_gen._logpdf.<locals>.f3S  s     2r
?"r2   rH  rL  	r?   rk   r  r
  rM  r  r  r  rD  s	            r0   r   zkappa4_gen._logpdf8  s    NN16162NN16162NN16162NN161624
	F	H	?	# 8B+q!9#%66+ 	+r2   c                 N    t        j                  | j                  |||            S rH   r  rO  s       r0   ro   zkappa4_gen._cdf^  r  r2   c                 >   t        j                  |dk7  |dk7        t        j                  |dk(  |dk7        t        j                  |dk7  |dk(        t        j                  |dk(  |dk(        g}d }d }d }d }t        |||||g|||gt         j                        S )Nr   c                 X    d|z  t        j                  | d|| z  z
  d|z  z  z        z  S )zVcdf = (1.0 - h*(1.0 - k*x)**(1.0/k))**(1.0/h)
               logcdf = ...
            r   r   rR  s      r0   r  zkappa4_gen._logcdf.<locals>.f0g  s4     E288QBac	SU';$;<<<r2   c                      d|| z  z
  d|z  z   S )zLcdf = np.exp(-(1.0 - k*x)**(1.0/k))
               logcdf = ...
            r   r   rR  s      r0   r  zkappa4_gen._logcdf.<locals>.f1m  s     1Q3Y#a%(((r2   c                 h    d|z  t        j                  | t        j                  |        z        z  S )zLcdf = (1.0 - h*np.exp(-x))**(1.0/h)
               logcdf = ...
            r   )rq   r  rJ   r   rR  s      r0   r  zkappa4_gen._logcdf.<locals>.f2s  s,     E288QBrvvqbzM222r2   c                 0    t        j                  |         S )zBcdf = np.exp(-np.exp(-x))
               logcdf = ...
            r.  rR  s      r0   rD  zkappa4_gen._logcdf.<locals>.f3y  s     FFA2J;r2   rH  rL  rV  s	            r0   r   zkappa4_gen._logcdfa  s    NN16162NN16162NN16162NN161624
	=	)	3	 8B+q!9#%66+ 	+r2   c                 >   t        j                  |dk7  |dk7        t        j                  |dk(  |dk7        t        j                  |dk7  |dk(        t        j                  |dk(  |dk(        g}d }d }d }d }t        |||||g|||gt         j                        S )Nr   c                 0    d|z  dd| |z  z
  |z  |z  z
  z  S r  r   rw   r  r
  s      r0   r  zkappa4_gen._ppf.<locals>.f0  s(    q5##A,!1A 5566r2   c                 F    d|z  dt        j                  |        |z  z
  z  S r  r+  r_  s      r0   r  zkappa4_gen._ppf.<locals>.f1  s$    q5#"&&)a/00r2   c                 b    t        j                  | |z          t        j                  |      z   S )z,ppf = -np.log((1.0 - (q**h))/h)
            r  r_  s      r0   r  zkappa4_gen._ppf.<locals>.f2  s)     HHq!tW%%q	11r2   c                 V    t        j                  t        j                  |               S rH   r+  r_  s      r0   rD  zkappa4_gen._ppf.<locals>.f3  s    FFBFF1I:&&&r2   rH  rL  )	r?   rw   r  r
  rM  r  r  r  rD  s	            r0   rx   zkappa4_gen._ppf  s    NN16162NN16162NN16162NN161624
	7	1	2
	' 8B+q!9#%66+ 	+r2   c                 v    t        j                  |dk  |dk\        |dk  g}d }d }t        |||g||gd      S )Nr   c                 8    d| z  |z  j                  t              S r6  astyper  r?  s     r0   r  z&kappa4_gen._get_stats_info.<locals>.f0  s    F1H$$S))r2   c                 2    d|z  j                  t              S r6  re  r?  s     r0   r  z&kappa4_gen._get_stats_info.<locals>.f1  s    F??3''r2   r:  rH  )rJ   r8  r   )r?   r  r
  rM  r  r  s         r0   _get_stats_infozkappa4_gen._get_stats_info  sK    NN1q5!q&)E

	*	( 8b"X1vqAAr2   c                     | j                  ||      }t        dd      D cg c],  }t        j                  ||k        rd nt        j                  . }}|d d  S c c}w Nr   r:  )rh  r  rJ   r  r  )r?   r  r
  maxrr  outputss         r0   r   zkappa4_gen._stats  sU    ##Aq)AFq!MA266!d(+47MMqz Ns   1Ac                     | j                  |d   |d         }||k\  rt        j                  S t        j                  | j
                  dd|f|z         d   S Nr   r   rF  )rh  rJ   r  r   rW  _mom_integ1)r?   r  rA   rk  s       r0   _mom1_sczkappa4_gen._mom1_sc  sR    ##DGT!W5966M~~d..1A49EaHHr2   N)r}   r~   r   r   r]   re   r   rl   r   ro   r   rx   rh  r   rp  r   r2   r0   r6  r6    sE    Wp/
'R-
$+L-!+F+2B
Ir2   r6  kappa4c                   L     e Zd ZdZd Zd Zd Z fdZd Zd Z	d Z
d	 Z xZS )

kappa3_gena*  Kappa 3 parameter distribution.

    %(before_notes)s

    Notes
    -----
    The probability density function for `kappa3` is:

    .. math::

        f(x, a) = a (a + x^a)^{-(a + 1)/a}

    for :math:`x > 0` and :math:`a > 0`.

    `kappa3` takes ``a`` as a shape parameter for :math:`a`.

    References
    ----------
    P.W. Mielke and E.S. Johnson, "Three-Parameter Kappa Distribution Maximum
    Likelihood and Likelihood Ratio Tests", Methods in Weather Research,
    701-707, (September, 1973),
    :doi:`10.1175/1520-0493(1973)101<0701:TKDMLE>2.3.CO;2`

    B. Kumphon, "Maximum Entropy and Maximum Likelihood Estimation for the
    Three-Parameter Kappa Distribution", Open Journal of Statistics, vol 2,
    415-419 (2012), :doi:`10.4236/ojs.2012.24050`

    %(after_notes)s

    %(example)s

    c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zkappa3_gen._shape_info  r  r2   c                 *    ||||z  z   d|z  dz
  z  z  S r2  r   r  s      r0   rl   zkappa3_gen._pdf  s"    !ad(d1fQh'''r2   c                 $    ||||z  z   d|z  z  z  S r6  r   r  s      r0   ro   zkappa3_gen._cdf  s    !ad(d1f%%%r2   c           	         t        j                  ||      \  }}t        |   ||      }d}||k  }t	        j
                  t	        j                  d||   z  ||   ||   ||    z  z               }||kD  }||   |   ||<   |||<   |S )Ng{Gz?r3  )rJ   r  r;   rs   rq   r  rq  )	r?   rk   r   sfcutoffr  sf2i2r  s	           r0   rs   zkappa3_gen._sf  s    ""1a(1W[A
 Kxx

4!A$;!qtadU{0BCDD6\Q%)B1	r2   c                 &    ||| z  dz
  z  d|z  z  S r  r   r  s      r0   rx   zkappa3_gen._ppf  s    1qb53;3q5))r2   c                 r    t        j                  | |       }t        j                  |      }||z  d|z  z  S r  r7  )r?   rw   r   lgr  s        r0   r{   zkappa3_gen._isf  s6    ZZQBE	S1W%%r2   c                     t        dd      D cg c],  }t        j                  ||k        rd nt        j                  . }}|d d  S c c}w rj  )r  rJ   r  r  )r?   r   r  rl  s       r0   r   zkappa3_gen._stats  sB    >CAqkJ266!a%=4bff4JJqz Ks   1Ac                     t        j                  ||d   k\        rt         j                  S t        j                  | j
                  dd|f|z         d   S rn  )rJ   r  r  r   rW  ro  )r?   r  rA   s      r0   rp  zkappa3_gen._mom1_sc  sE    66!tAw,66M~~d..1A49EaHHr2   )r}   r~   r   r   re   rl   ro   rs   rx   r{   r   rp  r  r  s   @r0   rs  rs    s3    @E(& *&
Ir2   rs  kappa3c                   B    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 Zy)	moyal_gena  A Moyal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `moyal` is:

    .. math::

        f(x) = \exp(-(x + \exp(-x))/2) / \sqrt{2\pi}

    for a real number :math:`x`.

    %(after_notes)s

    This distribution has utility in high-energy physics and radiation
    detection. It describes the energy loss of a charged relativistic
    particle due to ionization of the medium [1]_. It also provides an
    approximation for the Landau distribution. For an in depth description
    see [2]_. For additional description, see [3]_.

    References
    ----------
    .. [1] J.E. Moyal, "XXX. Theory of ionization fluctuations",
           The London, Edinburgh, and Dublin Philosophical Magazine
           and Journal of Science, vol 46, 263-280, (1955).
           :doi:`10.1080/14786440308521076` (gated)
    .. [2] G. Cordeiro et al., "The beta Moyal: a useful skew distribution",
           International Journal of Research and Reviews in Applied Sciences,
           vol 10, 171-192, (2012).
           http://www.arpapress.com/Volumes/Vol10Issue2/IJRRAS_10_2_02.pdf
    .. [3] C. Walck, "Handbook on Statistical Distributions for
           Experimentalists; International Report SUF-PFY/96-01", Chapter 26,
           University of Stockholm: Stockholm, Sweden, (2007).
           http://www.stat.rice.edu/~dobelman/textfiles/DistributionsHandbook.pdf

    .. versionadded:: 1.1.0

    %(example)s

    c                     g S rH   r   rd   s    r0   re   zmoyal_gen._shape_info5  r   r2   Nc                 `    t         j                  dd||      }t        j                  |       S )Nr   rO   )r   r*   r   r   )r  r  rJ   r   )r?   r   r   r  s       r0   r   zmoyal_gen._rvs8  s.    YYAD$0  2r
{r2   c                     t        j                  d|t        j                  |       z   z        t        j                  dt         j                  z        z  S Nr  rO   )rJ   r   r   r   r   s     r0   rl   zmoyal_gen._pdf=  s:    vvda"&&!*n-.2551AAAr2   c                     t        j                  t        j                  d|z        t        j                  d      z        S r  )rq   ro  rJ   r   r   r   s     r0   ro   zmoyal_gen._cdf@  s+    wwrvvdQh'"''!*455r2   c                     t        j                  t        j                  d|z        t        j                  d      z        S r  )rq   r  rJ   r   r   r   s     r0   rs   zmoyal_gen._sfC  s+    vvbffTAX&344r2   c                 `    t        j                  dt        j                  |      dz  z         S r  )rJ   r   rq   erfcinvr   s     r0   rx   zmoyal_gen._ppfF  s&    q2::a=!++,,,r2   c                    t        j                  d      t         j                  z   }t         j                  dz  dz  }dt        j                  d      z  t        j                  d      z  t         j                  dz  z  }d}||||fS )NrO      r  rA  )rJ   r   euler_gammar   r   rq   rO  r;  s        r0   r   zmoyal_gen._statsI  sg    VVAY'eeQhl"''!*_rwwqz)BEE1H43Br2   c                    |dk(  r&t        j                  d      t         j                  z   S |dk(  r@t         j                  dz  dz  t        j                  d      t         j                  z   dz  z   S |dk(  rdt         j                  dz  z  t        j                  d      t         j                  z   z  }t        j                  d      t         j                  z   dz  }dt	        j
                  d      z  }||z   |z   S |dk(  rd	t	        j
                  d      z  t        j                  d      t         j                  z   z  }dt         j                  dz  z  t        j                  d      t         j                  z   dz  z  }t        j                  d      t         j                  z   d
z  }dt         j                  d
z  z  d
z  }||z   |z   |z   S | j                  |      S )Nr   rO   r   r;  r  r  r  rA  8   r  r  )rJ   r   r  r   rq   rO  rp  )r?   r\   tmp1r0  tmp3tmp4s         r0   r   zmoyal_gen._munpP  sn   866!9r~~--#X55!8a<266!9r~~#="AAA#X>RVVAYr~~%=>DFF1Ibnn,q0D
?D$;%%#XBGGAJ&"&&)bnn*DEDruuax<266!9r~~#="AADFF1I.2Druuax<!#D$;%,, ==##r2   r   )r}   r~   r   r   re   r   rl   ro   rs   rx   r   r   r   r2   r0   r  r  
  s1    )T
B65-$r2   r  moyalc                   \    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zd ZddZddZy)nakagami_gena`  A Nakagami continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `nakagami` is:

    .. math::

        f(x, \nu) = \frac{2 \nu^\nu}{\Gamma(\nu)} x^{2\nu-1} \exp(-\nu x^2)

    for :math:`x >= 0`, :math:`\nu > 0`. The distribution was introduced in
    [2]_, see also [1]_ for further information.

    `nakagami` takes ``nu`` as a shape parameter for :math:`\nu`.

    %(after_notes)s

    References
    ----------
    .. [1] "Nakagami distribution", Wikipedia
           https://en.wikipedia.org/wiki/Nakagami_distribution
    .. [2] M. Nakagami, "The m-distribution - A general formula of intensity
           distribution of rapid fading", Statistical methods in radio wave
           propagation, Pergamon Press, 1960, 3-36.
           :doi:`10.1016/B978-0-08-009306-2.50005-4`

    %(example)s

    c                     |dkD  S r  r   )r?   nus     r0   r]   znakagami_gen._argcheck  r  r2   c                 @    t        dddt        j                  fd      gS )Nr  Fr   r  rb   rd   s    r0   re   znakagami_gen._shape_info  r  r2   c                 L    t        j                  | j                  ||            S rH   r  r?   rk   r  s      r0   rl   znakagami_gen._pdf  rV  r2   c                     t        j                  d      t        j                  ||      z   t        j                  |      z
  t        j                  d|z  dz
  |      z   ||dz  z  z
  S r  )rJ   r   rq   rr  r  r  s      r0   r   znakagami_gen._logpdf  s\     q	BHHR,,rzz"~=21%&(*1a40 	1r2   c                 :    t        j                  |||z  |z        S rH   r  r  s      r0   ro   znakagami_gen._cdf  s    {{2r!tAv&&r2   c                 `    t        j                  d|z  t        j                  ||      z        S r  r  )r?   rw   r  s      r0   rx   znakagami_gen._ppf  s%    wws2vbnnR3344r2   c                 :    t        j                  |||z  |z        S rH   r  r  s      r0   rs   znakagami_gen._sf  s    ||B1Q''r2   c                 `    t        j                  d|z  t        j                  ||      z        S rX   r  )r?   r  r  s      r0   r{   znakagami_gen._isf  s%    wwqtboob!4455r2   c                 (   t        j                  |d      t        j                  |      z  }d||z  z
  }|dd|z  |z  z
  z  dz  |z  t        j                  |d      z  }d|dz  z  |z  d|z  d	z
  |d	z  z  z   d	|z  z
  dz   }|||dz  z  z  }||||fS )
Nr   r   r   r  r   r  r'  rO   )rq   r  rJ   r   r  )r?   r  r<  r=  r>  r?  s         r0   r   znakagami_gen._stats  s    WWRbggbk)"R%i1qtCx< 3&+bhhsC.@@AXb[AbDFBE>)!B$.2
bck3Br2   c                    t        j                  |      }t        j                  |      }t        j                  |      }||dz
  t        j
                  |      z  z
  }dt        j                  |      z  t        j                  d      z
  }||z   |z   }t        j                  j                         }|dkD  }||   |z   dd||   z  z  z
  ||<   |j                  |      d   S )Nr   r  rO   g     j@r   r  r   )rJ   rn  r  rq   r  r  r   r  r   r   rE  )	r?   r  rn  r!  r"  r  r  norm_entropyr  s	            r0   r   znakagami_gen._entropy  s    ]]2JJrN"s(bjjn,,266":q	)EAIzz**, Htl"Q2a5\1!yy##r2   Nc                 T    t        j                  |j                  ||      |z        S r  )rJ   r   r  )r?   r  r   r   s       r0   r   znakagami_gen._rvs  s&    ww|222D2ABFGGr2   c                    t        |t              r|j                         }|d| j                  z  }t	        j
                  |      }t	        j                  t	        j                  ||z
  dz        t        |      z        }|||fz   S )N)r   rO   )	r9   r%   r  numargsrJ   r@  r   r  r  )r?   r@   rA   r)   r*   s        r0   r  znakagami_gen._fitstart  sq    dL)>>#D<DLL(D ffTls
Q/#d);<sEl""r2   r   rH   )r}   r~   r   r   r]   re   rl   r   ro   rx   rs   r{   r   r   r   r  r   r2   r0   r  r  i  sE    >F+1'5(6$"H	#r2   r  nakagamic                 6   |dz  dz
  }t        j                  |       t        j                  |      }}t        j                  |dz  | |z        d||z
  dz  z  z
  }t        j                  |||z        dz  }t        |dkD  ||fd t         j                         S )Nr   r   r   rO   r   c                 2    | t        j                  |      z   S rH   r+  )r  r  s     r0   r  z_ncx2_log_pdf.<locals>.<lambda>  s    q266!9} r2   )r|  r  )rJ   r   rq   rr  iver   rc   )rk   r  rH  df2r  nsr  corrs           r0   _ncx2_log_pdfr    s     S&3,CWWQZB
((3s7AbD
!Cb1$4
4C66#r"u#Dq	d
$66'	 r2   c                   N    e Zd ZdZd Zd ZddZd Zd Zd Z	d	 Z
d
 Zd Zd Zy)ncx2_gena  A non-central chi-squared continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `ncx2` is:

    .. math::

        f(x, k, \lambda) = \frac{1}{2} \exp(-(\lambda+x)/2)
            (x/\lambda)^{(k-2)/4}  I_{(k-2)/2}(\sqrt{\lambda x})

    for :math:`x >= 0`, :math:`k > 0` and :math:`\lambda \ge 0`.
    :math:`k` specifies the degrees of freedom (denoted ``df`` in the
    implementation) and :math:`\lambda` is the non-centrality parameter
    (denoted ``nc`` in the implementation). :math:`I_\nu` denotes the
    modified Bessel function of first order of degree :math:`\nu`
    (`scipy.special.iv`).

    `ncx2` takes ``df`` and ``nc`` as shape parameters.

    %(after_notes)s

    %(example)s

    c                 D    |dkD  t        j                  |      z  |dk\  z  S r  rY  r?   r  rH  s      r0   r]   zncx2_gen._argcheck  s"    Q"++b/)R1W55r2   c                     t        dddt        j                  fd      }t        dddt        j                  fd      }||gS )Nr  Fr   r  rH  ra   rb   r?   idfincs      r0   re   zncx2_gen._shape_info   s<    uq"&&k>Buq"&&k=ASzr2   Nc                 (    |j                  |||      S rH   )noncentral_chisquare)r?   r  rH  r   r   s        r0   r   zncx2_gen._rvs  s    00R>>r2   c                 r    t        j                  |t              |dk7  z  }t        ||||ft        d       S )Nr  r   c                 .    t         j                  | |      S rH   )r  r   rk   r  _s      r0   r  z"ncx2_gen._logpdf.<locals>.<lambda>  s    dll1b.A r2   r&  )rJ   	ones_likeboolr   r  r?   rk   r  rH  conds        r0   r   zncx2_gen._logpdf  s9    ||AT*bAg6$B}AC 	Cr2   c                     t        j                  |t              |dk7  z  }t        j                  d      5  t	        ||||ft
        j                  d       cd d d        S # 1 sw Y   y xY w)Nr  r   r2  rj  c                 .    t         j                  | |      S rH   )r  rl   r  s      r0   r  zncx2_gen._pdf.<locals>.<lambda>      $))Ar2B r2   r&  )rJ   r  r  r5  r   rl  	_ncx2_pdfr  s        r0   rl   zncx2_gen._pdf  ^    ||AT*bAg6[[h' 	DdQBK63C3C!BD	D 	D 	D   !A##A,c                     t        j                  |t              |dk7  z  }t        j                  d      5  t	        ||||ft
        j                  d       cd d d        S # 1 sw Y   y xY w)Nr  r   r2  rj  c                 .    t         j                  | |      S rH   )r  ro   r  s      r0   r  zncx2_gen._cdf.<locals>.<lambda>  r  r2   r&  )rJ   r  r  r5  r   rl  	_ncx2_cdfr  s        r0   ro   zncx2_gen._cdf  r  r  c                     t        j                  |t              |dk7  z  }t        j                  d      5  t	        ||||ft
        j                  d       cd d d        S # 1 sw Y   y xY w)Nr  r   r2  rj  c                 .    t         j                  | |      S rH   )r  rx   r  s      r0   r  zncx2_gen._ppf.<locals>.<lambda>  r  r2   r&  )rJ   r  r  r5  r   rl  	_ncx2_ppf)r?   rw   r  rH  r  s        r0   rx   zncx2_gen._ppf  r  r  c                     t        j                  |t              |dk7  z  }t        j                  d      5  t	        ||||ft
        j                  d       cd d d        S # 1 sw Y   y xY w)Nr  r   r2  rj  c                 .    t         j                  | |      S rH   )r  rs   r  s      r0   r  zncx2_gen._sf.<locals>.<lambda>#  s    $((1b/ r2   r&  )rJ   r  r  r5  r   rl  _ncx2_sfr  s        r0   rs   zncx2_gen._sf  s\    ||AT*bAg6[[h' 	CdQBK6??!AC	C 	C 	Cr  c                     t        j                  |t              |dk7  z  }t        j                  d      5  t	        ||||ft
        j                  d       cd d d        S # 1 sw Y   y xY w)Nr  r   r2  rj  c                 .    t         j                  | |      S rH   )r  r{   r  s      r0   r  zncx2_gen._isf.<locals>.<lambda>)  r  r2   r&  )rJ   r  r  r5  r   rl  	_ncx2_isfr  s        r0   r{   zncx2_gen._isf%  r  r  c                     t        j                  ||      t        j                  ||      t        j                  ||      t        j                  ||      fS rH   )rl  
_ncx2_mean_ncx2_variance_ncx2_skewness_ncx2_kurtosis_excessr  s      r0   r   zncx2_gen._stats+  sL    b"%!!"b)!!"b)((R0	
 	
r2   r   )r}   r~   r   r   r]   re   r   r   rl   ro   rx   rs   r{   r   r   r2   r0   r  r    s?    66
?C
DDDCD
r2   r  ncx2c                   P    e Zd ZdZd Zd ZddZd Zd Zd Z	d	 Z
d
 Zd ZddZy)ncf_gena  A non-central F distribution continuous random variable.

    %(before_notes)s

    See Also
    --------
    scipy.stats.f : Fisher distribution

    Notes
    -----
    The probability density function for `ncf` is:

    .. math::

        f(x, n_1, n_2, \lambda) =
            \exp\left(\frac{\lambda}{2} +
                      \lambda n_1 \frac{x}{2(n_1 x + n_2)}
                \right)
            n_1^{n_1/2} n_2^{n_2/2} x^{n_1/2 - 1} \\
            (n_2 + n_1 x)^{-(n_1 + n_2)/2}
            \gamma(n_1/2) \gamma(1 + n_2/2) \\
            \frac{L^{\frac{n_1}{2}-1}_{n_2/2}
                \left(-\lambda n_1 \frac{x}{2(n_1 x + n_2)}\right)}
            {B(n_1/2, n_2/2)
                \gamma\left(\frac{n_1 + n_2}{2}\right)}

    for :math:`n_1, n_2 > 0`, :math:`\lambda \ge 0`.  Here :math:`n_1` is the
    degrees of freedom in the numerator, :math:`n_2` the degrees of freedom in
    the denominator, :math:`\lambda` the non-centrality parameter,
    :math:`\gamma` is the logarithm of the Gamma function, :math:`L_n^k` is a
    generalized Laguerre polynomial and :math:`B` is the beta function.

    `ncf` takes ``df1``, ``df2`` and ``nc`` as shape parameters. If ``nc=0``,
    the distribution becomes equivalent to the Fisher distribution.

    %(after_notes)s

    %(example)s

    c                 $    |dkD  |dkD  z  |dk\  z  S r  r   )r?   df1r  rH  s       r0   r]   zncf_gen._argcheck`  s    aC!G$a00r2   c                     t        dddt        j                  fd      }t        dddt        j                  fd      }t        dddt        j                  fd      }|||gS )Nr  Fr   r  r  rH  ra   rb   )r?   idf1idf2r  s       r0   re   zncf_gen._shape_infoc  sW    %BFF^D%BFF^Duq"&&k=AdC  r2   Nc                 *    |j                  ||||      S rH   )noncentral_f)r?   r  r  rH  r   r   s         r0   r   zncf_gen._rvsi  s    ((c2t<<r2   c                 2    t        j                  ||||      S rH   )rl  _ncf_pdfr?   rk   r  r  rH  s        r0   rl   zncf_gen._pdfl  s     q#sB//r2   c                 2    t        j                  ||||      S rH   )rl  _ncf_cdfr  s        r0   ro   zncf_gen._cdfu  s    q#sB//r2   c                     t        j                  d      5  t        j                  ||||      cd d d        S # 1 sw Y   y xY wri  )rJ   r5  rl  _ncf_ppf)r?   rw   r  r  rH  s        r0   rx   zncf_gen._ppfx  5    [[h' 	4??1c33	4 	4 	4	   9Ac                 2    t        j                  ||||      S rH   )rl  _ncf_sfr  s        r0   rs   zncf_gen._sf|  s    ~~ac2..r2   c                     t        j                  d      5  t        j                  ||||      cd d d        S # 1 sw Y   y xY wri  )rJ   r5  rl  _ncf_isfr  s        r0   r{   zncf_gen._isf  r  r  c                 F   |dz  |z  |z  }t        j                  |d|z  z         t        j                  d|z  |z
        z   t        j                  |dz        z
  }|t        j                  | dz  |z         z  }|t        j                  |d|z  z   d|z  d|z        z  }|S )Nr   r   r   )rq   r  rJ   r   hyp1f1)r?   r\   r  r  rH  r7  r  s          r0   r   zncf_gen._munp  s    Sy}q zz!CG)$rzz#c'!)'<<rzz#c'?RRrvvrcCin%%ryy3s7CGSV44
r2   c                     t        j                  |||      }t        j                  |||      }d|v rt        j                  |||      nd }d|v rt        j                  |||      nd }||||fS Nr  r
  )rl  	_ncf_mean_ncf_variance_ncf_skewness_ncf_kurtosis_excess)	r?   r  r  rH  r  r<  r=  r>  r?  s	            r0   r   zncf_gen._stats  sz    c3+""3R036'>V!!#sB/t G^ ((b15 	3Br2   r   r  )r}   r~   r   r   r]   re   r   rl   ro   rx   rs   r{   r   r   r   r2   r0   r  r  7  s:    'P1!=004/4r2   r  ncfc                   N    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 Zd Zd Zy)t_gena  A Student's t continuous random variable.

    For the noncentral t distribution, see `nct`.

    %(before_notes)s

    See Also
    --------
    nct

    Notes
    -----
    The probability density function for `t` is:

    .. math::

        f(x, \nu) = \frac{\Gamma((\nu+1)/2)}
                        {\sqrt{\pi \nu} \Gamma(\nu/2)}
                    (1+x^2/\nu)^{-(\nu+1)/2}

    where :math:`x` is a real number and the degrees of freedom parameter
    :math:`\nu` (denoted ``df`` in the implementation) satisfies
    :math:`\nu > 0`. :math:`\Gamma` is the gamma function
    (`scipy.special.gamma`).

    %(after_notes)s

    %(example)s

    c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zt_gen._shape_info  r  r2   Nc                 (    |j                  ||      S r  )
standard_tr  s       r0   r   z
t_gen._rvs  s    &&r&55r2   c                 P     t        |t        j                  k(  ||fd  fd      S )Nc                 ,    t         j                  |       S rH   )r   rl   rk   r  s     r0   r  zt_gen._pdf.<locals>.<lambda>  s    DIIaL r2   c                 N    t        j                  j                  | |            S rH   r  )rk   r  r?   s     r0   r  zt_gen._pdf.<locals>.<lambda>  s    t||Ar*+ r2   r&  r  r  s   `  r0   rl   z
t_gen._pdf  s)    "&&L1b'(
 	
r2   c                 R    d }d }t        |t        j                  k(  ||f||      S )Nc                    t        j                  t        j                  d|z  d            dt        j                  |      t        j                  t         j                        z   z  z
  |dz   dz  t        j
                  | | z  |z        z  z
  S r  )rJ   r   rq   r  r   r  r 	  s     r0   t_logpdfzt_gen._logpdf.<locals>.t_logpdf  sn    FF27738S12RVVBZ"&&-789Avqj!a%(!334 5r2   c                 ,    t         j                  |       S rH   )r   r   r 	  s     r0   norm_logpdfz"t_gen._logpdf.<locals>.norm_logpdf  s    <<?"r2   r&  r  )r?   rk   r  r	  r	  s        r0   r   zt_gen._logpdf  s+    	5
	# ",B	[XNNr2   c                 .    t        j                  ||      S rH   rq   stdtrr  s      r0   ro   z
t_gen._cdf  r  r2   c                 0    t        j                  ||       S rH   r	  r  s      r0   rs   z	t_gen._sf  s    xxQBr2   c                 .    t        j                  ||      S rH   rq   stdtritr  s      r0   rx   z
t_gen._ppf  s    zz"a  r2   c                 0    t        j                  ||       S rH   r	  r  s      r0   r{   z
t_gen._isf  s    

2q!!!r2   c                    t        j                  |      }t        j                  |dkD  dt         j                        }|dkD  |dk  z  |dkD  t        j                  |      z  |f}d d d f}t        |||ft         j                        }t        j                  |dkD  dt         j                        }|dkD  |dk  z  |dkD  t        j                  |      z  |f}d	 d
 d f}t        |||ft         j                        }||||fS )Nr   r   rO   c                 ^    t        j                  t         j                  | j                        S rH   rJ   broadcast_torc   rn  r  s    r0   r  zt_gen._stats.<locals>.<lambda>      !B r2   c                     | | dz
  z  S r7  r   r  s    r0   r  zt_gen._stats.<locals>.<lambda>  s    r#v r2   c                 B    t        j                  d| j                        S rX   rJ   r	  rn  r  s    r0   r  zt_gen._stats.<locals>.<lambda>      BHH!= r2   r  r  c                 ^    t        j                  t         j                  | j                        S rH   r	  r  s    r0   r  zt_gen._stats.<locals>.<lambda>  r	  r2   c                     d| dz
  z  S )Nr:  rA  r   r  s    r0   r  zt_gen._stats.<locals>.<lambda>  s    3 r2   c                 B    t        j                  d| j                        S r  r	  r  s    r0   r  zt_gen._stats.<locals>.<lambda>  r	  r2   )rJ   isposinfrF  rc   r   r   r  )	r?   r  infinite_dfr<  rM  
choicelistr=  r>  r?  s	            r0   r   zt_gen._stats  s   kk"oXXb1fc266*!Va(!Vr{{2.! C.=?
 (Jrvv>XXb1fc266*!Va(!Vr{{2.! C/=?
 :ubff=3Br2   c                     |t         j                  k(  rt        j                         S d }d }t	        |dk\  |f||      }|S )Nc                     | dz  }| dz   dz  }|t        j                  |      t        j                  |      z
  z  t        j                  t        j                  |       t        j
                  |d      z        z   S r  )rq   r  rJ   r   r   rg  )r  halfhalf1s      r0   r  zt_gen._entropy.<locals>.regular  se    a4D!VQJE2::e,rzz$/??@ffRWWR[s);;<= >r2   c                     t         j                         d| z  z   | dz  dz  z   | dz  dz  z
  | dz  dz  z
  d| d	z  z  z   | d
z  dz  z   }|S )Nr   r  r  r  r  r  r'  g333333?r  r  )r   r   )r  r  s     r0   r$  z"t_gen._entropy.<locals>.asymptotic   sg     1R4'2s7A+5S!CGQ;!%r3w035s7A+>AHr2   d   r&  )rJ   rc   r   r   r   )r?   r  r  r$  r  s        r0   r   zt_gen._entropy  s@    <==?"	>	 rSy2&J7Cr2   r   r  r   r2   r0   r  r    s;    <F6

O !"4r2   r  r  c                   J    e Zd ZdZd Zd ZddZd Zd Zd Z	d	 Z
d
 ZddZy)nct_gena  A non-central Student's t continuous random variable.

    %(before_notes)s

    Notes
    -----
    If :math:`Y` is a standard normal random variable and :math:`V` is
    an independent chi-square random variable (`chi2`) with :math:`k` degrees
    of freedom, then

    .. math::

        X = \frac{Y + c}{\sqrt{V/k}}

    has a non-central Student's t distribution on the real line.
    The degrees of freedom parameter :math:`k` (denoted ``df`` in the
    implementation) satisfies :math:`k > 0` and the noncentrality parameter
    :math:`c` (denoted ``nc`` in the implementation) is a real number.

    %(after_notes)s

    %(example)s

    c                     |dkD  ||k(  z  S r  r   r  s      r0   r]   znct_gen._argcheck(  s    Q28$$r2   c                     t        dddt        j                  fd      }t        ddt        j                   t        j                  fd      }||gS )Nr  Fr   r  rH  rb   r  s      r0   re   znct_gen._shape_info+  sC    uq"&&k>Buw&7HSzr2   Nc                     t         j                  |||      }t        j                  |||      }|t        j                  |      z  t        j                  |      z  S )Nr  r  )r   r  r  rJ   r   )r?   r  rH  r   r   r\   r  s          r0   r   znct_gen._rvs0  sI    HH$\HBXXbt,X?2772;,,r2   c                 b   |dz  }|dz  }||z  }||z  |z  }||z   }|dz  t        j                  |      z  t        j                  |dz         z   |t        j                  d      z  ||z  dz  z   |dz  t        j                  |      z  z   t        j                  |dz        z   z
  }t        j                  |      }	|d|z  z  }
t        j
                  d      |z  |z  t        j                  |dz  dz   d|
      z  t        j                  |t        j                  |dz   dz        z        z  }t        j                  |dz   dz  d|
      t        j                  t        j
                  |      t        j                  |dz  dz         z        z  }|	||z   z  }	t        j                  |	dd       S )Nr   r   r   rO   r  r   r   )
rJ   r   rq   r  r   r   r  r   r  clip)r?   rk   r  rH  r\   r  r  r  trm1r  valFtrm2s               r0   rl   znct_gen._pdf5  s~   sFVqS"uRx2v"RVVAYAaC0RVVAY;Bq(AaC+==ZZ!_%& VVD\qv
2a		!A#a%d ;;**T"((AaC7"3345		1Q3'3-**RWWT]288AaCE?:;<
d4iwwr1d##r2   c                     t        j                  d      5  t        j                  t        j                  |||      dd      cd d d        S # 1 sw Y   y xY wNr2  rj  r   r   )rJ   r5  r*	  rl  _nct_cdfr?   rk   r  rH  s       r0   ro   znct_gen._cdfH  s@    [[h' 	=776??1b"5q!<	= 	= 	=   ,AAc                     t        j                  d      5  t        j                  |||      cd d d        S # 1 sw Y   y xY wri  )rJ   r5  rl  _nct_ppf)r?   rw   r  rH  s       r0   rx   znct_gen._ppfL  3    [[h' 	.??1b"-	. 	. 	.ro  c                     t        j                  d      5  t        j                  t        j                  |||      dd      cd d d        S # 1 sw Y   y xY wr/	  )rJ   r5  r*	  rl  _nct_sfr1	  s       r0   rs   znct_gen._sfP  s@    [[h' 	<776>>!R4a;	< 	< 	<r2	  c                     t        j                  d      5  t        j                  |||      cd d d        S # 1 sw Y   y xY wri  )rJ   r5  rl  _nct_isfr1	  s       r0   r{   znct_gen._isfT  r5	  ro  c                     t        j                  ||      }t        j                  ||      }d|v rt        j                  ||      nd }d|v rt        j                  ||      nd }||||fS r  )rl  	_nct_mean_nct_variance_nct_skewness_nct_kurtosis_excess)r?   r  rH  r  r<  r=  r>  r?  s           r0   r   znct_gen._statsX  sh    b"%""2r*-0G^V!!"b)477NV((R03Br2   r   r  )r}   r~   r   r   r]   re   r   rl   ro   rx   rs   r{   r   r   r2   r0   r%	  r%	    s4    0%
-
$&=.<.r2   r%	  nctc                   t     e Zd ZdZd Zd Zd Zd Zd Zd Z	ddZ
d	 Ze ee       fd
              Z xZS )
pareto_genaL  A Pareto continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `pareto` is:

    .. math::

        f(x, b) = \frac{b}{x^{b+1}}

    for :math:`x \ge 1`, :math:`b > 0`.

    `pareto` takes ``b`` as a shape parameter for :math:`b`.

    %(after_notes)s

    %(example)s

    c                 @    t        dddt        j                  fd      gS ru  rb   rd   s    r0   re   zpareto_gen._shape_infoy  r  r2   c                     ||| dz
  z  z  S rX   r   rw  s      r0   rl   zpareto_gen._pdf|  s    1r!t9}r2   c                     d|| z  z
  S rX   r   rw  s      r0   ro   zpareto_gen._cdf  s    1r7{r2   c                 &    t        d|z
  d|z        S )Nr   r3  r  r  s      r0   rx   zpareto_gen._ppf  s    1Q3Qr2   c                     || z  S rH   r   rw  s      r0   rs   zpareto_gen._sf  s    A2wr2   c                 4    t        j                  |d|z        S r6  r  r  s      r0   r{   zpareto_gen._isf  s    xx4!8$$r2   c                    d\  }}}}d|v rp|dkD  }t        j                  ||      }t        j                  t        j                  |      t         j                        }t        j
                  ||||dz
  z         d|v ry|dkD  }t        j                  ||      }t        j                  t        j                  |      t         j                        }t        j
                  ||||dz
  z  |dz
  dz  z         d	|v r|d
kD  }t        j                  ||      }t        j                  t        j                  |      t         j                        }d|dz   z  t        j                  |dz
        z  |dz
  t        j                  |      z  z  }	t        j
                  |||	       d|v r|dkD  }t        j                  ||      }t        j                  t        j                  |      t         j                        }dt        j                  g d|      z  t        j                  g d|      z  }	t        j
                  |||	       ||||fS )Nr+  r  r   r8  r   r  rO   r   r  r  r;  r
  r  r:  )r   r   r  r	  )r   g      r  r   )	rJ   extractr:  rn  rc   placer  r   r  )
r?   r   r  r<  r=  r>  r?  maskbtr  s
             r0   r   zpareto_gen._stats  s   0CR'>q5DD!$B!8BHHRrRV}-'>q5DD!$B''"((1+"&&9CHHS$bfC! ;<'>q5DD!$B!8BS>BGGBH$55"s(bggbk9QRDHHRt$'>q5DD!$B!8B

#5r::JJ5r:;DHHRt$3Br2   c                 >    dd|z  z   t        j                  |      z
  S rz  r+  rr  s     r0   r   zpareto_gen._entropy      3q5y266!9$$r2   c                    t        | ||      }|\  }}|;t        j                        |z
  |xs dk  rt        ddt        j                        j
                  d   fd||cxu r9n n5fdfdfdfd	}t        |j                  d
d            }|dz  |dz  }
}	 ||	|
      sE|	dkD  s|
t        j                  k  r-|	dz  }	|
dz  }
 ||	|
      s|	dkD  r|
t        j                  k  r-t        |	|
g      }|j                  r|j                  }t        j                        |z
  }xs	  ||      }||z   t        j                        k  s.t        j                        |z
  }t        j                  |d      }|||fS t        | 4  fi |S |t        j                        |z
  }n|}|xs t        j                        |z
  }xs	  ||      }|||fS )Nr   paretor   r  c                 f    t        j                  t        j                  |z
  | z              z  S rH   r  )r*   locationr@   ndatas     r0   	get_shapez!pareto_gen.fit.<locals>.get_shape  s+     266"&&$/U)B"CDDDr2   c                     | z  |z  S rH   r   )rn  r*   rS	  s     r0   	dL_dScalez!pareto_gen.fit.<locals>.dL_dScale  s     u}u,,r2   c                 F    | dz   t        j                  d|z
  z        z  S rX   r  )rn  rR	  r@   s     r0   dL_dLocationz$pareto_gen.fit.<locals>.dL_dLocation  s&     	RVVA,A%BBBr2   c                 t    t        j                        | z
  }xs	  | |      } ||       ||       z
  S rH   )rJ   r@  )r*   rR	  rn  rX	  rV	  r@   r  rT	  s      r0   r  z$pareto_gen.fit.<locals>.fun_to_solve  sA     66$<%/<)E8"<#E84y7NNNr2   c                 r    t        j                   |             t        j                   |            k7  S rH   rI   rL   rM   r  s     r0   rN   z.pareto_gen.fit.<locals>.interval_contains_root  s/    V 45V 456 7r2   r*   rO   r  )r  rJ   r@  rD  rc   rn  rA  r7   r&   r  r  r  r;   r=   )r?   r@   rA   r/   r  r   r   rN   r  rL   rM   r  r*   r)   rn  rX	  rV	  r  r  rT	  rS	  r  s    `             @@@@@@r0   r=   zpareto_gen.fit  s    1tT4H
%/"fdF tt 3v{ Cxq??

1	E
 6!!-
C
O O7  ! 45K(1_kAoFF .ff=
frvvo!! .ff=
frvvo lVV4DEC}}ffTlU*7)E3"7 rvvd|3FF4L3.ELL2Ec5((w{40400\&&,'CC ,"&&,,/)E3/c5  r2   r  )r}   r~   r   r   re   rl   ro   rx   rs   r{   r   r   rE   r
   r   r=   r  r  s   @r0   rA	  rA	  c  sT    *E %6% M*R! + R!r2   rA	  rP	  c                   L    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zd Zy)	lomax_gena  A Lomax (Pareto of the second kind) continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `lomax` is:

    .. math::

        f(x, c) = \frac{c}{(1+x)^{c+1}}

    for :math:`x \ge 0`, :math:`c > 0`.

    `lomax` takes ``c`` as a shape parameter for :math:`c`.

    `lomax` is a special case of `pareto` with ``loc=-1.0``.

    %(after_notes)s

    %(example)s

    c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zlomax_gen._shape_info  r  r2   c                 $    |dz  d|z   |dz   z  z  S r  r   r  s      r0   rl   zlomax_gen._pdf  s    uc!equ%%%r2   c                 d    t        j                  |      |dz   t        j                  |      z  z
  S rX   r  r  s      r0   r   zlomax_gen._logpdf#  s&    vvayAaC!,,,r2   c                 \    t        j                  | t        j                  |      z         S rH   r  r  s      r0   ro   zlomax_gen._cdf&  s"    !BHHQK(((r2   c                 Z    t        j                  | t        j                  |      z        S rH   )rJ   r   rq   r  r  s      r0   rs   zlomax_gen._sf)  s    vvqb!n%%r2   c                 4    | t        j                  |      z  S rH   r   r  s      r0   r   zlomax_gen._logsf,  s    r"((1+~r2   c                 \    t        j                  t        j                  |        |z        S rH   r  r  s      r0   rx   zlomax_gen._ppf/  s!    xx1"a((r2   c                     |d|z  z  dz
  S r2  r   r  s      r0   r{   zlomax_gen._isf2  s    4!8}q  r2   c                 H    t         j                  |dd      \  }}}}||||fS )Nr3  rC  )r)   r  )rP	  r  rP  s         r0   r   zlomax_gen._stats5  s,     ,,qdF,CCR3Br2   c                 >    dd|z  z   t        j                  |      z
  S rz  r+  rr  s     r0   r   zlomax_gen._entropy9  s    Qwrvvay  r2   N)r}   r~   r   r   re   rl   r   ro   rs   r   rx   r{   r   r   r   r2   r0   r]	  r]	    s:    .E&-)&)!!r2   r]	  lomaxc                        e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zdd
Zd Ze eed       fd              Z xZS )pearson3_gena  A pearson type III continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `pearson3` is:

    .. math::

        f(x, \kappa) = \frac{|\beta|}{\Gamma(\alpha)}
                       (\beta (x - \zeta))^{\alpha - 1}
                       \exp(-\beta (x - \zeta))

    where:

    .. math::

            \beta = \frac{2}{\kappa}

            \alpha = \beta^2 = \frac{4}{\kappa^2}

            \zeta = -\frac{\alpha}{\beta} = -\beta

    :math:`\Gamma` is the gamma function (`scipy.special.gamma`).
    Pass the skew :math:`\kappa` into `pearson3` as the shape parameter
    ``skew``.

    %(after_notes)s

    %(example)s

    References
    ----------
    R.W. Vogel and D.E. McMartin, "Probability Plot Goodness-of-Fit and
    Skewness Estimation Procedures for the Pearson Type 3 Distribution", Water
    Resources Research, Vol.27, 3149-3158 (1991).

    L.R. Salvosa, "Tables of Pearson's Type III Function", Ann. Math. Statist.,
    Vol.1, 191-198 (1930).

    "Using Modern Computing Tools to Fit the Pearson Type III Distribution to
    Aviation Loads Data", Office of Aviation Research (2003).

    c                     d}d}d}t        j                  d||      \  }}}|j                         }t        j                  |      |k  }| }d||   |z  z  }	||	z  dz  }
||
|	z  z
  }|	||   |z
  z  }||||||	|
|fS )Nr   r   g>r   rO   )rJ   r  copyr  )r?   rk   r   r)   r*   norm2pearson_transitionansrK	  invmaskrg  r  rO  transxs                r0   _preprocesszpearson3_gen._preprocessn  s    
  #+**348Qhhj {{4 #::%d7me+,!UT\!7d*+AvtWdE4??r2   c                 ,    t        j                  |      S rH   rY  )r?   r   s     r0   r]   zpearson3_gen._argcheck  s    
 {{4  r2   c                 ^    t        ddt        j                   t        j                  fd      gS )Nr   Fr  rb   rd   s    r0   re   zpearson3_gen._shape_info  s%    65BFF7BFF*;^LMMr2   c                 *    d}d}|}d|dz  z  }||||fS )Nr   r   r  rO   r   )r?   r   r  r  r  r
  s         r0   r   zpearson3_gen._stats  s,    aK!Qzr2   c                     t        j                  | j                  ||            }|j                  dk(  rt        j                  |      ry|S d|t        j                  |      <   |S )Nr   r   )rJ   r   r   r"  rX  )r?   rk   r   rn	  s       r0   rl   zpearson3_gen._pdf  sR    
 ffT\\!T*+88q=xx}J BHHSM
r2   c                     | j                  ||      \  }}}}}}}}	t        j                  t        ||               ||<   t        j                  t	        |            t
        j                  ||      z   ||<   |S rH   )rq	  rJ   r   r   r  r  r  )
r?   rk   r   rn	  rp	  rK	  ro	  rg  r  r  s
             r0   r   zpearson3_gen._logpdf  ss     Q% 	6QgtUA FF9QtW-.D	 vvc$i(5<<+FFG
r2   c                    | j                  ||      \  }}}}}}}}t        ||         ||<   t        j                  ||j                        }t        j
                  ||dkD        }	||   dkD  }
t        j                  ||
   ||
         ||	<   t        j
                  ||dk        }||   dk  }t        j                  ||   ||         ||<   |S r  )	rq	  r   rJ   r	  rn  r8  r  r   rx  r?   rk   r   rn	  rp	  rK	  ro	  r  r  	invmask1a	invmask1b	invmask2a	invmask2bs                r0   ro   zpearson3_gen._cdf  s    Q% 	3Qgq% ag&D	tW]]3NN7D1H5	MA%	 6)#4eI6FGI NN7D1H5	MA%	&"3U95EFI
r2   c                    | j                  ||      \  }}}}}}}}t        ||         ||<   t        j                  ||j                        }t        j
                  ||dkD        }	||   dkD  }
t        j                  ||
   ||
         ||	<   t        j
                  ||dk        }||   dk  }t        j                  ||   ||         ||<   |S r  )	rq	  r   rJ   r	  rn  r8  r  rx  r   rx	  s                r0   rs   zpearson3_gen._sf  s    Q% 	3Qgq% QtW%D	tW]]3NN7D1H5	MA%	&"3U95EFINN7D1H5	MA%	6)#4eI6FGI
r2   c                    t        j                  ||      }| j                  dg|      \  }}}}}}}	}
|j                         }|j                  |z
  }|j                  |      ||<   |j                  |	|      |z  |
z   ||<   |dk(  r|d   }|S )Nr   r   )rJ   r	  rq	  r  r   r   r  )r?   r   r   r   rn	  r  rK	  ro	  rg  r  rO  nsmallnbigs                r0   r   zpearson3_gen._rvs  s    tT*aS$' 	4Q4$t yy6! 008D	#225$?DtKG2:a&C
r2   c                     | j                  ||      \  }}}}}}}}	t        ||         ||<   ||   }d||dk     z
  ||dk  <   t        j                  ||      |z  |	z   ||<   |S r@  )rq	  r   rq   r  )
r?   rw   r   rn	  r  rK	  ro	  rg  r  rO  s
             r0   rx   zpearson3_gen._ppf  s~    Q% 	4Q4$tag&D	gJ!D1H+o$(~~eQ/4t;G
r2   ze        Note that method of moments (`method='MM'`) is not
        available for this distribution.

r   c                 |    |j                  dd       dk(  rt        d      t        t        |       |   |g|i |S )Nr,   MMzhFit `method='MM'` is not available for the Pearson3 distribution. Please try the default `method='MLE'`.)r7   NotImplementedErrorr;   r<   r=   r  s       r0   r=   zpearson3_gen.fit  sO    
 88Hd#t+% 'D E E dT.tCdCdCCr2   r   )r}   r~   r   r   rq	  r]   re   r   rl   r   ro   rs   r   rx   rE   r   r   r=   r  r  s   @r0   rj	  rj	  @  sg    ,Z@8!N0" } 50 1D1 Dr2   rj	  pearson3c                        e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zd Z fdZe eed       fd              Z xZS )powerlaw_gena  A power-function continuous random variable.

    %(before_notes)s

    See Also
    --------
    pareto

    Notes
    -----
    The probability density function for `powerlaw` is:

    .. math::

        f(x, a) = a x^{a-1}

    for :math:`0 \le x \le 1`, :math:`a > 0`.

    `powerlaw` takes ``a`` as a shape parameter for :math:`a`.

    %(after_notes)s

    For example, the support of `powerlaw` can be adjusted from the default
    interval ``[0, 1]`` to the interval ``[c, c+d]`` by setting ``loc=c`` and
    ``scale=d``. For a power-law distribution with infinite support, see
    `pareto`.

    `powerlaw` is a special case of `beta` with ``b=1``.

    %(example)s

    c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zpowerlaw_gen._shape_info(   r  r2   c                     |||dz
  z  z  S r  r   r  s      r0   rl   zpowerlaw_gen._pdf+   s    QsU|r2   c                 `    t        j                  |      t        j                  |dz
  |      z   S rX   )rJ   r   rq   rr  r  s      r0   r   zpowerlaw_gen._logpdf/   s$    vvay288AE1---r2   c                     ||dz  z  S r  r   r  s      r0   ro   zpowerlaw_gen._cdf2   s    1S5zr2   c                 2    |t        j                  |      z  S rH   r+  r  s      r0   r   zpowerlaw_gen._logcdf5   r,  r2   c                      t        |d|z        S r  r  r  s      r0   rx   zpowerlaw_gen._ppf8   s    1c!e}r2   c                 0    t        j                  ||       S rH   )rq   rG  )r?   r  r   s      r0   rs   zpowerlaw_gen._sf;   s    Ar2   c                     |||z   z  S rH   r   )r?   r\   r   s      r0   r   zpowerlaw_gen._munp>   s    AE{r2   c                     ||dz   z  ||dz   z  |dz   dz  z  d|dz
  |dz   z  z  t        j                  |dz   |z        z  dt        j                  g d|      z  ||dz   z  |dz   z  z  fS )	Nr   r   rO   r  r;  r  )r   r  r  rO   r  )rJ   r   r  r  s     r0   r   zpowerlaw_gen._statsB   s    QWQWSQ.SQW-.!c'Q1GGBJJ~q11Q!c']a!e5LMO 	Or2   c                 >    dd|z  z
  t        j                  |      z
  S rz  r+  r  s     r0   r   zpowerlaw_gen._entropyH   rN	  r2   c                 <    t         |   ||      |dk7  |dk\  z  z  S Nr   r   )r;   r  )r?   rk   r   r  s      r0   r  zpowerlaw_gen._support_maskK   s,    %a+FqAv&( 	)r2   a:          Notes specifically for ``powerlaw.fit``: If the location is a free
        parameter and the value returned for the shape parameter is less than
        one, the true maximum likelihood approaches infinity. This causes
        numerical difficulties, and the resulting estimates are approximate.
        

r   c                 R   |j                  dd      rt        |   g|i |S t        t	        j
                              dk(  rt        |   g|i |S t        | ||      \  }}| j                        fg}| j                  |i       d   }|Ej                         |kD  st        ddd      |#j                         ||z   k  st        ddd      |5|dk  rt        d      |t	        j                        k  rd}t        |      d d	 || ||      ||fS |t	        j                  j                         t        j                         }	xs
  |	|      }
 ||
|	|f      }t	        j                  j                         |z
  t        j                        }xs
  ||      } ||||f      }||k  r|
|	|fS |||fS | |      }xs
  ||      }|||fS fd
}d d fdfdfd}dk  r |       S dkD  r |       S  |       }| j!                  |      } |       }| j!                  |      }||k  r
|d   dk  r|S ||kD  r
|d   dkD  r|S t        |   g|i |S )Nr  Fr   powerlawr   zKNegative or zero `fscale` is outside the range allowed by the distribution.z0`fscale` must be greater than the range of data.c                     t        |       }| t        j                  t        j                  | |z
              |t        j                  |      z  z
  z  S rH   )r  rJ   r  r   )r@   r)   r*   r  s       r0   rT	  z#powerlaw_gen.fit.<locals>.get_shape   sA     D	A3"&&s
!34qFGGr2   c                 (    | j                         |z
  S rH   )rY  )r@   r)   s     r0   	get_scalez#powerlaw_gen.fit.<locals>.get_scale   s     88:##r2   c                     t        j                  j                         t         j                         } t        j                  |       t        j
                  | j                        j                  k  r?t        j                  |       t        j
                  | j                        j                  z  } t        j                   |       t         j                        }xs
  | |      }|| |fS rH   )	rJ   r  r@  rc   r  r  r  r  rK   )r)   r*   rn  r@   r  r	  rT	  s      r0   fit_loc_scale_w_shape_lt_1z4powerlaw_gen.fit.<locals>.fit_loc_scale_w_shape_lt_1   s    ,,txxzBFF73Cvvc{RXXcii0555ggclRXXcii%8%=%==LL4!5rvv>E9ic59E#u$$r2   c                 .    | j                   d    |z  |z  S r  )rn  )r@   rn  r*   s      r0   rV	  z#powerlaw_gen.fit.<locals>.dL_dScale   s     JJqM>E)E11r2   c                 D    |dz
  t        j                  d|| z
  z        z  S rX   r  )r@   rn  r)   s      r0   rX	  z&powerlaw_gen.fit.<locals>.dL_dLocation   s%     AIS4Z(8!999r2   c                     t        j                   |       t         j                         }xs
  | |      } ||       S rH   rJ   r  rc   )r)   r*   rn  rX	  r@   r  r	  rT	  s      r0   dL_dLocation_starz+powerlaw_gen.fit.<locals>.dL_dLocation_star   sD     LL4!5w?E9ic59EeS11r2   c                     t        j                   |       t         j                         }xs
  | |      } ||       ||       z
  S rH   r	  )	r)   r*   rn  rX	  rV	  r@   r  r	  rT	  s	      r0   r  z&powerlaw_gen.fit.<locals>.fun_to_solve   sW     LL4!5w?E9ic59EdE51"445 6r2   c                     t        j                  
j                         t         j                         } 
j                         | z
  } 	|       dkD  r$
j                         |z
  } |dz  } 	|       dkD  r$fd}| dz
  }d} |||       sJ|t         j                   k7  r6
j                         |z
  }|dz  } |||       s|t         j                   k7  r6t	        j
                  || f      }t        j                  |j                  t         j                         }t        j                   
|      t         j                        }xs
  
||      }|||fS )Nr   rO   c                 r    t        j                   |             t        j                   |            k7  S rH   rI   r[	  s     r0   rN   zTpowerlaw_gen.fit.<locals>.fit_loc_scale_w_shape_gt_1.<locals>.interval_contains_root   s/    V 4577<#789 :r2   r   r   r  )rJ   r  r@  rc   r   r&   r  )rM   r  rN   rL   r  r  r)   r*   rn  r	  r@   r  r  r	  rT	  s            r0   fit_loc_scale_w_shape_gt_1z4powerlaw_gen.fit.<locals>.fit_loc_scale_w_shape_gt_1   s7    \\$((*rvvg6F XXZ&(E#F+a/e+
 $F+a/:
 aZF
 A-ff="&&(((*q.Q .ff="&&( ''vv>NOD,,tyy266'2CLL4!5rvv>E9ic59E#u$$r2   )r-   r;   r=   r  rJ   uniquer  r  _reduce_funcr@  rD  rY  r   ptpr  rc   r  )r?   r@   rA   r/   r   r   penalized_nllf_argspenalized_nllfrS   loc_lt1	shape_lt1ll_lt1loc_gt1	shape_gt1ll_gt1r*   rn  r	  r	  fit_shape_lt1fit_shape_gt1rX	  r	  rV	  r  r  r	  rT	  r  s    `                   @@@@@@@r0   r=   zpowerlaw_gen.fitO   s    P 88J&7;t3d3d33ryy1$7;t3d3d33%@tAEt&M"fdF#dnnT&:%<=**+>CAF
 88:$":q!44!$((*v*E":q!44{  "F G G%H o%	H	$ $"2T40$>> ll488:w7GB)D'6"BI#Y$@$GF ll488:#6?GB)D'6"BI#Y$@$GF '611 '611 dD)E:idE:E$%%
	%	2
	:
	2 	2	6 	6!	% !	%H &A+-//FQJ-// 34=$/24=$/Va 0A 5  f_q!1A!5  7;t3d3d33r2   )r}   r~   r   r   re   rl   r   ro   r   rx   rs   r   r   r   r  rE   r   r   r=   r  r  s   @r0   r	  r	     sl    @E.O%) } 5 H4 H4r2   r	  r	  c                   X    e Zd ZdZej
                  Zd Zd Zd Z	d Z
d Zd Zd Zd	 Zy
)powerlognorm_gena  A power log-normal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `powerlognorm` is:

    .. math::

        f(x, c, s) = \frac{c}{x s} \phi(\log(x)/s)
                     (\Phi(-\log(x)/s))^{c-1}

    where :math:`\phi` is the normal pdf, and :math:`\Phi` is the normal cdf,
    and :math:`x > 0`, :math:`s, c > 0`.

    `powerlognorm` takes :math:`c` and :math:`s` as shape parameters.

    %(after_notes)s

    %(example)s

    c                     t        dddt        j                  fd      }t        dddt        j                  fd      }||gS )Nr  Fr   r  r  rb   )r?   r  r*  s      r0   re   zpowerlognorm_gen._shape_info>!  r+  r2   c                 N    t        j                  | j                  |||            S rH   r  r?   rk   r  r  s       r0   rl   zpowerlognorm_gen._pdfC!  r  r2   c                    t        j                  |      t        j                  |      z
  t        j                  |      z
  t        t        j                  |      |z        z   t        t        j                  |       |z        |dz
  z  z   S r  rJ   r   r   r   r	  s       r0   r   zpowerlognorm_gen._logpdfF!  si    q	BFF1I%q	1RVVAY]+,bffQiZ!^,B78 	9r2   c                 P    t        j                  | j                  |||             S rH   rX  r	  s       r0   ro   zpowerlognorm_gen._cdfK!  rY  r2   c                 .    | j                  d|z
  ||      S rX   )r{   r?   rw   r  r  s       r0   rx   zpowerlognorm_gen._ppfN!  s    yyQ1%%r2   c                 N    t        j                  | j                  |||            S rH   r-  r	  s       r0   rs   zpowerlognorm_gen._sfQ!  r.  r2   c                 L    t        t        j                  |       |z        |z  S rH   rN  r	  s       r0   r   zpowerlognorm_gen._logsfT!  s     RVVAYJN+a//r2   c                 R    t        j                  t        |d|z  z         |z        S rX   r  r	  s       r0   r{   zpowerlognorm_gen._isfW!  s&    vvyQqS**Q.//r2   N)r}   r~   r   r   r   r  r  re   rl   r   ro   rx   rs   r   r{   r   r2   r0   r	  r	  $!  s<    . "44M
-9
/&,00r2   r	  powerlognormc                   @    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zy
)powernorm_genah  A power normal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `powernorm` is:

    .. math::

        f(x, c) = c \phi(x) (\Phi(-x))^{c-1}

    where :math:`\phi` is the normal pdf, :math:`\Phi` is the normal cdf,
    :math:`x` is any real, and :math:`c > 0` [1]_.

    `powernorm` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    References
    ----------
    .. [1] NIST Engineering Statistics Handbook, Section 1.3.6.6.13,
           https://www.itl.nist.gov/div898/handbook//eda/section3/eda366d.htm

    %(example)s

    c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zpowernorm_gen._shape_infoz!  r  r2   c                 D    |t        |      z  t        |       |dz
  z  z  S r  r   r   r  s      r0   rl   zpowernorm_gen._pdf}!  s$    1~A23!788r2   c                 j    t        j                  |      t        |      z   |dz
  t        |       z  z   S rX   r	  r  s      r0   r   zpowernorm_gen._logpdf!  s.    vvay<?*ac<3C-CCCr2   c                 N    t        j                  | j                  ||             S rH   rX  r  s      r0   ro   zpowernorm_gen._cdf!  s    Q*+++r2   c                 :    t        t        d|z
  d|z               S r  )r   r  r  s      r0   rx   zpowernorm_gen._ppf!  s    #cAgsQw/000r2   c                 L    t        j                  | j                  ||            S rH   r-  r  s      r0   rs   zpowernorm_gen._sf!  r  r2   c                      |t        |       z  S rH   r   r  s      r0   r   zpowernorm_gen._logsf!  s    <###r2   c                 l    t        t        j                  t        j                  |      |z               S rH   )r   rJ   r   r   r  s      r0   r{   zpowernorm_gen._isf!  s%    "&&Q/000r2   N)r}   r~   r   r   re   rl   r   ro   rx   rs   r   r{   r   r2   r0   r	  r	  ^!  s1    6E9D,1)$1r2   r	  	powernormc                   B    e Zd ZdZd Zd Zd Zd Zd Zd Z	dd	Z
d
 Zy)	rdist_gena/  An R-distributed (symmetric beta) continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `rdist` is:

    .. math::

        f(x, c) = \frac{(1-x^2)^{c/2-1}}{B(1/2, c/2)}

    for :math:`-1 \le x \le 1`, :math:`c > 0`. `rdist` is also called the
    symmetric beta distribution: if B has a `beta` distribution with
    parameters (c/2, c/2), then X = 2*B - 1 follows a R-distribution with
    parameter c.

    `rdist` takes ``c`` as a shape parameter for :math:`c`.

    This distribution includes the following distribution kernels as
    special cases::

        c = 2:  uniform
        c = 3:  `semicircular`
        c = 4:  Epanechnikov (parabolic)
        c = 6:  quartic (biweight)
        c = 8:  triweight

    %(after_notes)s

    %(example)s

    c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zrdist_gen._shape_info!  r  r2   c                 L    t        j                  | j                  ||            S rH   r  r  s      r0   rl   zrdist_gen._pdf!  rK  r2   c                 v    t        j                  d       t        j                  |dz   dz  |dz  |dz        z   S r  )rJ   r   rg  r   r  s      r0   r   zrdist_gen._logpdf!  s4    q	zDLL!a%AaC1===r2   c                 H    t         j                  |dz   dz  |dz  |dz        S r*  r  r  s      r0   ro   zrdist_gen._cdf!  s%    yy!a%AaC1--r2   c                 H    t         j                  |dz   dz  |dz  |dz        S r*  r  r  s      r0   rs   zrdist_gen._sf!  s%    xxQ	1Q3!,,r2   c                 H    dt         j                  ||dz  |dz        z  dz
  S r  )rg  rx   r  s      r0   rx   zrdist_gen._ppf!  s'    1ac1Q3''!++r2   Nc                 @    d|j                  |dz  |dz  |      z  dz
  S r  rf  r  s       r0   r   zrdist_gen._rvs!  s)    <$$QqS!A#t44q88r2   c                     d|dz  z
  t        j                  |dz   dz  |dz        z  }|t        j                  d|dz        z  S )Nr   rO   r   r   r   rK  )r?   r\   r  	numerators       r0   r   zrdist_gen._munp!  sE    !a%[BGGQWM1s7$CC	27761r6222r2   r   )r}   r~   r   r   re   rl   r   ro   rs   rx   r   r   r   r2   r0   r	  r	  !  s1     BE*>.-,93r2   r	  r3  rdistc                        e Zd ZdZej
                  Zd ZddZd Z	d Z
d Zd Zd Zd	 Zd
 Zd Zd Ze eed       fd              Z xZS )rayleigh_gena7  A Rayleigh continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `rayleigh` is:

    .. math::

        f(x) = x \exp(-x^2/2)

    for :math:`x \ge 0`.

    `rayleigh` is a special case of `chi` with ``df=2``.

    %(after_notes)s

    %(example)s

    c                     g S rH   r   rd   s    r0   re   zrayleigh_gen._shape_info!  r   r2   c                 2    t         j                  d||      S )NrO   r  r  r   s      r0   r   zrayleigh_gen._rvs!  s    wwqt,w??r2   c                 J    t        j                  | j                  |            S rH   r  r?   r  s     r0   rl   zrayleigh_gen._pdf!  r  r2   c                 >    t        j                  |      d|z  |z  z
  S r  r+  r	  s     r0   r   zrayleigh_gen._logpdf!  s    vvay37Q;&&r2   c                 :    t        j                  d|dz  z         S r  r2  r	  s     r0   ro   zrayleigh_gen._cdf!  s    1%%%r2   c                 Z    t        j                  dt        j                  |       z        S Nr	  )rJ   r   rq   r  r   s     r0   rx   zrayleigh_gen._ppf!  s     wwrBHHaRL())r2   c                 J    t        j                  | j                  |            S rH   r-  r	  s     r0   rs   zrayleigh_gen._sf"  s    vvdkk!n%%r2   c                     d|z  |z  S )Nr  r   r	  s     r0   r   zrayleigh_gen._logsf"  s    ax!|r2   c                 X    t        j                  dt        j                  |      z        S r	  )rJ   r   r   r   s     r0   r{   zrayleigh_gen._isf"  s    wwrBFF1I~&&r2   c                 8   dt         j                  z
  }t        j                  t         j                  dz        |dz  dt         j                  dz
  z  t        j                  t         j                        z  |dz  z  dt         j                  z  |z  d|dz  z  z
  fS )Nr  rO   r  r  r  r%  r"  r#  s     r0   r   zrayleigh_gen._stats"  sy    "%%ia A2557BGGBEEN*383"%%BsAvI%' 	'r2   c                 L    t         dz  dz   dt        j                  d      z  z
  S )Nr   r   r   rO   r  rd   s    r0   r   zrayleigh_gen._entropy"  s!    czA~BFF1I--r2   a          Notes specifically for ``rayleigh.fit``: If the location is fixed with
        the `floc` parameter, this method uses an analytical formula to find
        the scale.  Otherwise, this function uses a numerical root finder on
        the first order conditions of the log-likelihood function to find the
        MLE.  Only the (optional) `loc` parameter is used as the initial guess
        for the root finder; the `scale` parameter and any other parameters
        for the optimizer are ignored.

r   c                    |j                  dd      rt        |   g|i |S t        | ||      \  }}fd}fd}|ffd	}|At	        j
                  |z
  dk        rt        ddt        j                  	      | ||      fS |j                  d
      }	|	| j                        d   }	||n|}
t	        j                  t	        j                        t        j                         }t        |
|      }t        j                  |
||f      }|j                  st!        |j"                        |j$                  }|xs  ||      }||fS )Nr  Fc                 ^    t        j                  | z
  dz        dt              z  z  dz  S r  )rJ   r  r  )r)   r@   s    r0   	scale_mlez#rayleigh_gen.fit.<locals>.scale_mle$"  s/     FFD3J1,-SY?BFFr2   c                     | z
  }|j                         }|dz  j                         }d|z  j                         }||dt              z  z  |z  z
  S r  )r  r  )r)   r  rU  r[  s3r@   s        r0   loc_mlez!rayleigh_gen.fit.<locals>.loc_mle)"  sT     BBa%BB$BAc$iK(+++r2   c                 b    | z
  }|j                         |dz  d|z  j                         z  z
  S r  )r  )r)   r*   r  r@   s      r0   loc_mle_scale_fixedz-rayleigh_gen.fit.<locals>.loc_mle_scale_fixed2"  s2     B668eQh!B$555r2   r   rayleighr   r  r)   r  )r-   r;   r=   r  rJ   r  rD  rc   r7   r  r  r@  rT   r   r&   r  rQ   flagr  )r?   r@   rA   r/   r   r   r	  r	  r	  loc0rC   rM   rL   r  r)   r*   r  s    `              r0   r=   zrayleigh_gen.fit"  sF    88J&7;t3d3d338t9=tEdF	G
	, ,2 	6 vvdTkQ&'":QbffEEYt_,, xx<>>$'*Dg-@bffTlRVVG4"3/""30@A}} **hh()C.Ezr2   r   )r}   r~   r   r   r   r  r  re   r   rl   r   ro   rx   rs   r   r{   r   r   rE   r   r=   r  r  s   @r0   r	  r	  !  su    * "44M@''&*&''. } 5. /// /r2   r	  r	  c                        e Zd ZdZd Zd Z fdZd Zd Zd Z	d Z
d	 Zd
 Zd ZdZ eee       fd       Z xZS )reciprocal_gena,  A loguniform or reciprocal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for this class is:

    .. math::

        f(x, a, b) = \frac{1}{x \log(b/a)}

    for :math:`a \le x \le b`, :math:`b > a > 0`. This class takes
    :math:`a` and :math:`b` as shape parameters.

    %(after_notes)s

    %(example)s

    This doesn't show the equal probability of ``0.01``, ``0.1`` and
    ``1``. This is best when the x-axis is log-scaled:

    >>> import numpy as np
    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots(1, 1)
    >>> ax.hist(np.log10(r))
    >>> ax.set_ylabel("Frequency")
    >>> ax.set_xlabel("Value of random variable")
    >>> ax.xaxis.set_major_locator(plt.FixedLocator([-2, -1, 0]))
    >>> ticks = ["$10^{{ {} }}$".format(i) for i in [-2, -1, 0]]
    >>> ax.set_xticklabels(ticks)  # doctest: +SKIP
    >>> plt.show()

    This random variable will be log-uniform regardless of the base chosen for
    ``a`` and ``b``. Let's specify with base ``2`` instead:

    >>> rvs = %(name)s(2**-2, 2**0).rvs(size=1000)

    Values of ``1/4``, ``1/2`` and ``1`` are equally likely with this random
    variable.  Here's the histogram:

    >>> fig, ax = plt.subplots(1, 1)
    >>> ax.hist(np.log2(rvs))
    >>> ax.set_ylabel("Frequency")
    >>> ax.set_xlabel("Value of random variable")
    >>> ax.xaxis.set_major_locator(plt.FixedLocator([-2, -1, 0]))
    >>> ticks = ["$2^{{ {} }}$".format(i) for i in [-2, -1, 0]]
    >>> ax.set_xticklabels(ticks)  # doctest: +SKIP
    >>> plt.show()

    c                     |dkD  ||kD  z  S r  r   r  s      r0   r]   zreciprocal_gen._argcheck"      A!a%  r2   c                     t        dddt        j                  fd      }t        dddt        j                  fd      }||gS r`  rb   ra  s      r0   re   zreciprocal_gen._shape_info"  rd  r2   c                     t        |t              r|j                         }t        |   |t        j                  |      t        j                  |      f      S NrF  r9   r%   r  r;   r  rJ   r@  rY  r  s     r0   r  zreciprocal_gen._fitstart"  sC    dL)>>#Dw RVVD\266$<,H IIr2   c                 
    ||fS rH   r   r  s      r0   r   zreciprocal_gen._get_support"  r  r2   c                 N    t        j                  | j                  |||            S rH   r  rn  s       r0   rl   zreciprocal_gen._pdf"  r  r2   c                     t        j                  |       t        j                  t        j                  |      t        j                  |      z
        z
  S rH   r+  rn  s       r0   r   zreciprocal_gen._logpdf"  s5    q	zBFF266!9rvvay#8999r2   c                     t        j                  |      t        j                  |      z
  t        j                  |      t        j                  |      z
  z  S rH   r+  rn  s       r0   ro   zreciprocal_gen._cdf"  s7    q	"&&)#q	BFF1I(=>>r2   c                     t        j                  t        j                  |      |t        j                  |      t        j                  |      z
  z  z         S rH   rJ   r   r   r}  s       r0   rx   zreciprocal_gen._ppf"  s8    vvbffQi!RVVAY%:";;<<r2   c                 *   dt        j                  |      t        j                  |      z
  z  |z  }t        j                  t        j                  t	        |t        j                  |      z  |t        j                  |      z                    }||z  S rX   )rJ   r   r  r   	_log_diff)r?   r\   r   r   r  r  s         r0   r   zreciprocal_gen._munp"  se    "&&)bffQi'(1,WWRVVIa"&&)mQrvvay[ABCBwr2   c                     dt        j                  |      t        j                  |      z   z  t        j                  t        j                  |      t        j                  |      z
        z   S r  r+  r  s      r0   r   zreciprocal_gen._entropy"  sE    BFF1Iq	)*RVVBFF1Iq	4I-JJJr2   z        `loguniform`/`reciprocal` is over-parameterized. `fit` automatically
         fixes `scale` to 1 unless `fscale` is provided by the user.

r   c                 R    |j                  dd      }t        |   |g|d|i|S )Nr   r   )r-   r;   r=   )r?   r@   rA   r/   r   r  s        r0   r=   zreciprocal_gen.fit"  s1    (A&w{4>$>v>>>r2   )r}   r~   r   r   r]   re   r  r   rl   r   ro   rx   r   r   fit_noter   r   r=   r  r  s   @r0   r	  r	  S"  s`    2f!
J-:?=
KLH }H=? >?r2   r	  
loguniform
reciprocalc                   <    e Zd ZdZd Zd Zd
dZd Zd Zd Z	d	 Z
y)rice_gena  A Rice continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `rice` is:

    .. math::

        f(x, b) = x \exp(- \frac{x^2 + b^2}{2}) I_0(x b)

    for :math:`x >= 0`, :math:`b > 0`. :math:`I_0` is the modified Bessel
    function of order zero (`scipy.special.i0`).

    `rice` takes ``b`` as a shape parameter for :math:`b`.

    %(after_notes)s

    The Rice distribution describes the length, :math:`r`, of a 2-D vector with
    components :math:`(U+u, V+v)`, where :math:`U, V` are constant, :math:`u,
    v` are independent Gaussian random variables with standard deviation
    :math:`s`.  Let :math:`R = \sqrt{U^2 + V^2}`. Then the pdf of :math:`r` is
    ``rice.pdf(x, R/s, scale=s)``.

    %(example)s

    c                     |dk\  S r  r   r?   r   s     r0   r]   zrice_gen._argcheck"  r  r2   c                 @    t        dddt        j                  fd      gS )Nr   Fr   ra   rb   rd   s    r0   re   zrice_gen._shape_info"  r  r2   Nc                     |t        j                  d      z  |j                  d|z         z   }t        j                  ||z  j                  d            S )NrO   )rO   r  r   r  )rJ   r   r   r  )r?   r   r   r   r  s        r0   r   zrice_gen._rvs"  sH    bggajL<77TD[7IIww!yyay())r2   c                 |    t        j                  t        j                  |      dt        j                  |            S r  )rq   chndtrrJ   r  rw  s      r0   ro   zrice_gen._cdf"  s%    yy1q"))A,77r2   c           	      |    t        j                  t        j                  |dt        j                  |                  S r  )rJ   r   rq   chndtrixr  r  s      r0   rx   zrice_gen._ppf"  s&    wwr{{1a1677r2   c                 ~    |t        j                  ||z
   ||z
  z  dz        z  t        j                  ||z        z  S r7  )rJ   r   rq   i0erw  s      r0   rl   zrice_gen._pdf"  s<     266AaC&!A#,s*++bffQqSk99r2   c                     |dz  }d|z   }||z  dz  }d|z  t        j                  |       z  t        j                  |      z  t        j                  |d|      z  S r  )rJ   r   rq   r  r  )r?   r\   r   nd2n1rq  s         r0   r   zrice_gen._munp"  s^    eWqSWc
RVVRC[(288B<7		"a$% 	&r2   r   )r}   r~   r   r   r]   re   r   ro   rx   rl   r   r   r2   r0   r
  r
  "  s+    8D*
88:&r2   r
  ricec                   6    e Zd ZdZd Zd Zd Zd Zd Zd	dZ	y)
recipinvgauss_gena  A reciprocal inverse Gaussian continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `recipinvgauss` is:

    .. math::

        f(x, \mu) = \frac{1}{\sqrt{2\pi x}}
                    \exp\left(\frac{-(1-\mu x)^2}{2\mu^2x}\right)

    for :math:`x \ge 0`.

    `recipinvgauss` takes ``mu`` as a shape parameter for :math:`\mu`.

    %(after_notes)s

    %(example)s

    c                 @    t        dddt        j                  fd      gS rH  rb   rd   s    r0   re   zrecipinvgauss_gen._shape_info#  r  r2   c                 L    t        j                  | j                  ||            S rH   r  rO  s      r0   rl   zrecipinvgauss_gen._pdf#  s     vvdll1b)**r2   c                 J    t        |dkD  ||fd t        j                         S )Nr   c                     d|| z  z
  dz   d| z  |dz  z  z  dt        j                  dt         j                  z  | z        z  z
  S )Nr   r   rO   r   r   )rk   r<  s     r0   r  z+recipinvgauss_gen._logpdf.<locals>.<lambda>"#  sI    1r!t8c/)9QqSS[)I+.rvvagai/@+@*A r2   r  r  rO  s      r0   r   zrecipinvgauss_gen._logpdf #  s*    !a%!RB%'VVG- 	-r2   c                     d|z  |z
  }d|z  |z   }dt        j                  |      z  }t        | |z        t        j                  d|z        t        | |z        z  z
  S Nr   r   rJ   r   r   r   r?   rk   r<  r+	  r-	  isqxs         r0   ro   zrecipinvgauss_gen._cdf&#  s_    2vz2vz2771:~$t$rvvc"f~id
6K'KKKr2   c                     d|z  |z
  }d|z  |z   }dt        j                  |      z  }t        ||z        t        j                  d|z        t        | |z        z  z   S r
  r
  r
  s         r0   rs   zrecipinvgauss_gen._sf,#  s]    2vz2vz2771:~d#bffSVnYuTz5J&JJJr2   Nc                 0    d|j                  |d|      z  S rJ  rK  rM  s       r0   r   zrecipinvgauss_gen._rvs2#  s    <$$R4$888r2   r   )
r}   r~   r   r   re   rl   r   ro   rs   r   r   r2   r0   r
  r
  #  s(    ,F+
-LK9r2   r
  recipinvgaussc                   B    e Zd ZdZd Zd Zd Zd Zd ZddZ	d	 Z
d
 Zy)semicircular_gena  A semicircular continuous random variable.

    %(before_notes)s

    See Also
    --------
    rdist

    Notes
    -----
    The probability density function for `semicircular` is:

    .. math::

        f(x) = \frac{2}{\pi} \sqrt{1-x^2}

    for :math:`-1 \le x \le 1`.

    The distribution is a special case of `rdist` with `c = 3`.

    %(after_notes)s

    References
    ----------
    .. [1] "Wigner semicircle distribution",
           https://en.wikipedia.org/wiki/Wigner_semicircle_distribution

    %(example)s

    c                     g S rH   r   rd   s    r0   re   zsemicircular_gen._shape_infoX#  r   r2   c                 `    dt         j                  z  t        j                  d||z  z
        z  S r  r"  r   s     r0   rl   zsemicircular_gen._pdf[#  s%    255y1Q3''r2   c                     t        j                  dt         j                  z        dt        j                  | |z        z  z   S r  r  r   s     r0   r   zsemicircular_gen._logpdf^#  s0    vvagRXXqbd^!333r2   c                     ddt         j                  z  |t        j                  d||z  z
        z  t        j                  |      z   z  z   S )Nr   r   r   )rJ   r   r   r#  r   s     r0   ro   zsemicircular_gen._cdfa#  s<    3ruu9a!A#.1=>>>r2   c                 .    t         j                  |d      S Nr  )r	  rx   r   s     r0   rx   zsemicircular_gen._ppfd#  s    zz!Qr2   Nc                     t        j                  |j                  |            }t        j                  t         j                  |j                  |      z        }||z  S r  )rJ   r   r  r  r   )r?   r   r   r  r   s        r0   r   zsemicircular_gen._rvsg#  sN     GGL((d(34FF255<//T/::;1ur2   c                      y)N)r   r  r   r3  r   rd   s    r0   r   zsemicircular_gen._statsn#  r%  r2   c                      y)NgzCϑ?r   rd   s    r0   r   zsemicircular_gen._entropyq#  s    %r2   r   )r}   r~   r   r   re   rl   r   ro   rx   r   r   r   r   r2   r0   r%
  r%
  9#  s/    <(4?  &r2   r%
  semicircularc                   <    e Zd ZdZd Zd Zd Zd Zd Zd
dZ	d Z
y	)skewcauchy_gena  A skewed Cauchy random variable.

    %(before_notes)s

    See Also
    --------
    cauchy : Cauchy distribution

    Notes
    -----

    The probability density function for `skewcauchy` is:

    .. math::

        f(x) = \frac{1}{\pi \left(\frac{x^2}{\left(a\, \text{sign}(x) + 1
                                                   \right)^2} + 1 \right)}

    for a real number :math:`x` and skewness parameter :math:`-1 < a < 1`.

    When :math:`a=0`, the distribution reduces to the usual Cauchy
    distribution.

    %(after_notes)s

    References
    ----------
    .. [1] "Skewed generalized *t* distribution", Wikipedia
       https://en.wikipedia.org/wiki/Skewed_generalized_t_distribution#Skewed_Cauchy_distribution

    %(example)s

    c                 2    t        j                  |      dk  S rX   )rJ   r  r  s     r0   r]   zskewcauchy_gen._argcheck#  s    vvay1}r2   c                      t        dddd      gS )Nr   F)r3  r   r  r   rd   s    r0   re   zskewcauchy_gen._shape_info#  s    3{NCDDr2   c                 x    dt         j                  |dz  |t        j                  |      z  dz   dz  z  dz   z  z  S r*  )rJ   r   rK   r  s      r0   rl   zskewcauchy_gen._pdf#  s:    BEEQTQ^a%7!$;;a?@AAr2   c                    t        j                  |dk  d|z
  dz  d|z
  t         j                  z  t        j                  |d|z
  z        z  z   d|z
  dz  d|z   t         j                  z  t        j                  |d|z   z        z  z         S Nr   r   rO   )rJ   rF  r   r}  r  s      r0   ro   zskewcauchy_gen._cdf#  s    xxQQ!q1uo		!q1u+8N&NNQ!q1uo		!q1u+8N&NNP 	Pr2   c           
      >   || j                  d|      k  }t        j                  |t        j                  t        j                  d|z
  z  |d|z
  dz  z
  z        d|z
  z  t        j                  t        j                  d|z   z  |d|z
  dz  z
  z        d|z   z        S r7
  )ro   rJ   rF  r  r   )r?   rk   r   r  s       r0   rx   zskewcauchy_gen._ppf#  s    		!QxxruuA!q1uk/BCq1uMruuA!q1uk/BCq1uMO 	Or2   c                 ~    t         j                  t         j                  t         j                  t         j                  fS rH   r  )r?   r   r  s      r0   r   zskewcauchy_gen._stats#  r  r2   c                     t        |t              r|j                         }t        j                  |g d      \  }}}d|||z
  dz  fS )Nr  r   rO   r  )r?   r@   r  r  r  s        r0   r  zskewcauchy_gen._fitstart#  sE     dL)>>#DdL9S#C#)Q&&r2   NrB  )r}   r~   r   r   r]   re   rl   ro   rx   r   r  r   r2   r0   r1
  r1
  x#  s/     BEBP
O.'r2   r1
  
skewcauchyc                        e Zd ZdZd Zd Zd Zd Z fdZd Z	d Z
d	 Zdd
ZddZed        Zd Z eed       fd       Z xZS )skewnorm_genaf  A skew-normal random variable.

    %(before_notes)s

    Notes
    -----
    The pdf is::

        skewnorm.pdf(x, a) = 2 * norm.pdf(x) * norm.cdf(a*x)

    `skewnorm` takes a real number :math:`a` as a skewness parameter
    When ``a = 0`` the distribution is identical to a normal distribution
    (`norm`). `rvs` implements the method of [1]_.

    %(after_notes)s

    %(example)s

    References
    ----------
    .. [1] A. Azzalini and A. Capitanio (1999). Statistical applications of
        the multivariate skew-normal distribution. J. Roy. Statist. Soc.,
        B 61, 579-602. :arxiv:`0911.2093`

    c                 ,    t        j                  |      S rH   rY  r  s     r0   r]   zskewnorm_gen._argcheck#  rZ  r2   c                 ^    t        ddt        j                   t        j                  fd      gS )Nr   Fr  rb   rd   s    r0   re   zskewnorm_gen._shape_info#  r]  r2   c                 .    t        |dk(  ||fd d       S )Nr   c                     t        |       S rH   r   rk   r   s     r0   r  z#skewnorm_gen._pdf.<locals>.<lambda>#  s
    1 r2   c                 <    dt        |       z  t        || z        z  S r7  r	  rB
  s     r0   r  z#skewnorm_gen._pdf.<locals>.<lambda>#  s    By|OIacN: r2   r  r  r  s      r0   rl   zskewnorm_gen._pdf#  s"    FQF5:
 	
r2   c                 .    t        |dk(  ||fd d       S )Nr   c                     t        |       S rH   r   rB
  s     r0   r  z&skewnorm_gen._logpdf.<locals>.<lambda>#  s
    a r2   c                 b    t        j                  d      t        |       z   t        || z        z   S r  r	  rB
  s     r0   r  z&skewnorm_gen._logpdf.<locals>.<lambda>#  s%    BFF1Il1o5l1Q36GG r2   r  r  r  s      r0   r   zskewnorm_gen._logpdf#  s"    FQF8G
 	
r2   c                    t        j                  |      }t        j                  |dd|      }t        j                  ||j
                        }|dk  |dkD  z  }t        |   ||   ||         ||<   t        j                  |dd      S )Nr   r   gư>)	rJ   r  rl  _skewnorm_cdfr	  rn  r;   ro   r*	  )r?   rk   r   r   i_small_cdfr  s        r0   ro   zskewnorm_gen._cdf#  s}    MM!""1aA.OOAsyy)Tza!e, 7<++GKwwsAq!!r2   c                 2    t        j                  |dd|      S r	  )rl  _skewnorm_ppfr  s      r0   rx   zskewnorm_gen._ppf#      ##Aq!Q//r2   c                 *    | j                  | |       S rH   r9  r  s      r0   rs   zskewnorm_gen._sf#  s     yy!aR  r2   c                 2    t        j                  |dd|      S r	  )rl  _skewnorm_isfr  s      r0   r{   zskewnorm_gen._isf#  rL
  r2   c                    |j                  |      }|j                  |      }|t        j                  d|dz  z         z  }||z  |t        j                  d|dz  z
        z  z   }t        j                  |dk\  ||       S )Nr  r   rO   r   )normalrJ   r   rF  )r?   r   r   r   u0r  r  r  s           r0   r   zskewnorm_gen._rvs#  s      d +T*bgga!Q$hrTAbgga!Q$h'''xxabS))r2   c                    g d}t        j                  dt         j                  z        |z  t        j                  d|dz  z         z  }d|v r||d<   d|v rd|dz  z
  |d<   d|v r;dt         j                  z
  dz  |t        j                  d|dz  z
        z  d	z  z  |d<   d
|v r+dt         j                  d	z
  z  |dz  d|dz  z
  dz  z  z  |d	<   |S )Nr+  rO   r   r  r   r  r  r  r  r
  r  )r?   r   r  r$  consts        r0   r   zskewnorm_gen._stats$  s    )"%% 1$RWWQAX%66'>F1I'>E1HF1I'>bee)Q5UAX1F+F*JJF1I'>BEEAI5!8Q\A4E+EFF1Ir2   c                     t        dg      t        ddg      t        g d      t        g d      t        g d      t        g d      t        g d      t        g d	      t        g d
      t        g d      d
}|S )Nr   r  r  )r  ir  )i   i?   i)i  iin  irV
  )(  iSi6Q  ii  iO)i iBi/ iio irX
  ) iԷi iYei{Hx ii i!)	i!iׅi쇀iiViX'ilirY
  )
is_'il   </1 ldy( l   J8D l.~ l   -Rx iWi[i0)
r   r  r:  r  r	  r     r        r   )r?   skewnorm_odd_momentss     r0   _skewnorm_odd_momentsz"skewnorm_gen._skewnorm_odd_moments$  s     1#1b'",'./78EF # $ 8 9 % &  2 3 
$ $#r2   c                    |dz  rP|dkD  rt        d      |t        j                  d|dz  z         z  }| | j                  |   |dz        z  t        z  S t        j                  |dz   dz        d|dz  z  z  t        z  S )Nr   r\
  zKskewnorm noncentral moments not implemented for odd orders greater than 19.rO   )r	  rJ   r   r^
  r#   rq   r  r"   )r?   r  r   r  s       r0   r   zskewnorm_gen._munp1$  s    19rz) +5 6 6
 bgga!Q$h''E=D66u=eQhGG%& ' 88UQYM*Qq\9HDDr2   a          If ``method='mm'``, parameters fixed by the user are respected, and the
        remaining parameters are used to match distribution and sample moments
        where possible. For example, if the user fixes the location with
        ``floc``, the parameters will only match the distribution skewness and
        variance to the sample skewness and variance; no attempt will be made
        to match the means or minimize a norm of the errors.
        Note that the maximum possible skewness magnitude of a
        `scipy.stats.skewnorm` distribution is approximately 0.9952717; if the
        magnitude of the data's sample skewness exceeds this, the returned
        shape parameter ``a`` will be infinite.
        

r   c           	      4   |j                  dd      rt        |   |g|i |S t        |t              r7|j                         dk(  r|j                         }nt        |   |g|i |S t        | |||      \  }}}}|j                  dd      j                         }d }d }	|dk(  rd	\  }
}}n6t        |      r|d   nd }
|j                  d
d       }|j                  dd       }||
t        j                  |      }|dk(  rt        j                  |dd      }n  |d      }t        j                  || |      } |	|      }t        j                  d      5  t        j                   t        j"                  |dz  d|dz  z
              t        j$                  |      z  }
d d d        n$||n|
}
|
t        j                   d|
dz  z         z  }|J|Ht        j&                  |      }t        j                   |dd|dz  z  t        j(                  z  z
  z        }n||}|G|Et        j*                  |      }|||z  t        j                   dt        j(                  z        z  z
  }n||}|dk(  r|
||fS t        |   ||
f||d|S # 1 sw Y   xY w)Nr  Fr   r,   r5   c                     dt         j                  z
  dz  | t        j                  dt         j                  z        z  dz  dd| dz  z  t         j                  z  z
  dz  z  z  S )Nr  rO   r  r   r  r"  r  s    r0   skew_dz skewnorm_gen.fit.<locals>.skew_db$  s]    beeGQ;1rwwq255y'9#9A"=%&1a4"%%%73$?#@ A Ar2   c                     t        j                  |       dz  }t        j                  |       t        j                  t         j                  dz  |z  |dt         j                  z
  dz  dz  z   z        z  S )NrA  rO   r  )rJ   r  rK   r   r   )r   s_23s     r0   d_skewz skewnorm_gen.fit.<locals>.d_skewe$  s^    66$<#&D774=277a$$1ruu9a-3)?"?@$  r2   r6   r  r)   r*   gGzgGz?r   r2  r3  rO   r  )r-   r;   r=   r9   r%   r:   r  r  r7   r8   r  r  r   rJ   r*	  r5  r   r4  rK   r  r   r   )r?   r@   rA   r/   r  r   r   r,   rc
  rf
  r   r)   r*   r  s_maxr  r  r  r  s                     r0   r=   zskewnorm_gen.fitE$  s    88J&7;t3d3d33dL)  "a'~~'w{47$7$77 "=T4=A4"Ib$(E*002	A	 T>,MAsEt9Q$A((5$'CHHWd+E:!) 

4 A GGAud+q	GGAvu-q	AH- BGGBIIadQq!tV56rwwqzAB B n!ABGGA1H%%A>emtAGGAQq!tVBEE\!123EE<CKAeAgbggag...CCT>c5=  7;tQECuEEE/B Bs   A	JJr   rB  )r}   r~   r   r   r]   re   rl   r   ro   rx   rs   r{   r   r   r   r^
  r   r   r   r=   r  r  s   @r0   r=
  r=
  #  sy    2K

"0!
0** $ $*E( } 5 FFFFr2   r=
  skewnormc                   :    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
y	)
trapezoid_gena  A trapezoidal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The trapezoidal distribution can be represented with an up-sloping line
    from ``loc`` to ``(loc + c*scale)``, then constant to ``(loc + d*scale)``
    and then downsloping from ``(loc + d*scale)`` to ``(loc+scale)``.  This
    defines the trapezoid base from ``loc`` to ``(loc+scale)`` and the flat
    top from ``c`` to ``d`` proportional to the position along the base
    with ``0 <= c <= d <= 1``.  When ``c=d``, this is equivalent to `triang`
    with the same values for `loc`, `scale` and `c`.
    The method of [1]_ is used for computing moments.

    `trapezoid` takes :math:`c` and :math:`d` as shape parameters.

    %(after_notes)s

    The standard form is in the range [0, 1] with c the mode.
    The location parameter shifts the start to `loc`.
    The scale parameter changes the width from 1 to `scale`.

    %(example)s

    References
    ----------
    .. [1] Kacker, R.N. and Lawrence, J.F. (2007). Trapezoidal and triangular
       distributions for Type B evaluation of standard uncertainty.
       Metrologia 44, 117-127. :doi:`10.1088/0026-1394/44/2/003`


    c                 <    |dk\  |dk  z  |dk\  z  |dk  z  ||k\  z  S r	  r   r?   r  r  s      r0   r]   ztrapezoid_gen._argcheck$  s0    Q16"a1f-a8AFCCr2   c                 B    t        dddd      }t        dddd      }||gS )Nr  Fr   r   TTr  r4
  r  s      r0   re   ztrapezoid_gen._shape_info$  s+    UHl;UHl;Bxr2   c                 j    d||z
  dz   z  }t        ||k  ||k  ||k  z  ||kD  gd d d g||||f      S )NrO   r   c                     || z  |z  S rH   r   rk   r  r  r  s       r0   r  z$trapezoid_gen._pdf.<locals>.<lambda>$  s    q1uqy r2   c                     |S rH   r   rr
  s       r0   r  z$trapezoid_gen._pdf.<locals>.<lambda>$  s    q r2   c                     |d| z
  z  d|z
  z  S rX   r   rr
  s       r0   r  z$trapezoid_gen._pdf.<locals>.<lambda>$  s    qAaCyAaC/@ r2   r   )r?   rk   r  r  r  s        r0   rl   ztrapezoid_gen._pdf$  s`    1QKAE!VQ/E# 90@B q!Q<) 	)r2   c                 R    t        ||k  ||k  ||k  z  ||kD  gd d d g|||f      S )Nc                 $    | dz  |z  ||z
  dz   z  S r  r   rk   r  r  s      r0   r  z$trapezoid_gen._cdf.<locals>.<lambda>$  s    AqD1H!A,> r2   c                 *    |d| |z
  z  z   ||z
  dz   z  S r  r   rx
  s      r0   r  z$trapezoid_gen._cdf.<locals>.<lambda>$  s    Qac]qs1u,E r2   c                 6    dd| z
  dz  ||z
  dz   z  d|z
  z  z
  S r*  r   rx
  s      r0   r  z$trapezoid_gen._cdf.<locals>.<lambda>$  s3    A!z23A#a%09<=aC0A -B r2   ru
  r*  s       r0   ro   ztrapezoid_gen._cdf$  sP    AE!VQ/E# ?EBC q!9& 	&r2   c                 R   | j                  |||      | j                  |||      }}||k  ||k  ||kD  g}t        j                  ||z  d|z   |z
  z        d|z  d|z   |z
  z  d|z  z   dt        j                  d|z
  ||z
  dz   z  d|z
  z        z
  g}t        j                  ||      S r  )ro   rJ   r   select)r?   rw   r  r  qcqdrM  r	  s           r0   rx   ztrapezoid_gen._ppf$  s    1a#TYYq!Q%7BFAGQV,gga!eq1uqy12AgQ+cAg5"''1q5QUQY"71q5"ABBD
 yy:..r2   c                     |dz   z  }t        |dk(  d|k  |dk  z  |dk(  gd fdfdg|g      }dd|z   |z
  z  ||z
  z  dz   dz   z  z  }|S )	Nr   r   r   c                      yr  r   rb
  s    r0   r  z%trapezoid_gen._munp.<locals>.<lambda>$  s    r2   c                 l    t        j                  dz   t        j                  |       z        | dz
  z  S rr  )rJ   r  r   r  r\   s    r0   r  z%trapezoid_gen._munp.<locals>.<lambda>$  s*    rxx1q	 12ae< r2   c                     dz   S r  r   r
  s    r0   r  z%trapezoid_gen._munp.<locals>.<lambda>$  s    qs r2   r   rO   ru
  )r?   r\   r  r  ab_termdc_termr7  s    `     r0   r   ztrapezoid_gen._munp$  s     ac(#XaAG,a3h7< C SU1Wo7!23!!}E
r2   c                 h    dd|z
  |z   z  d|z   |z
  z  t        j                  dd|z   |z
  z        z   S r   r+  rl
  s      r0   r   ztrapezoid_gen._entropy$  s=     c!eAg#a%'*RVVC3q57O-DDDr2   N)r}   r~   r   r   r]   re   rl   ro   rx   r   r   r   r2   r0   rj
  rj
  $  s-     BD
	)&/2Er2   rj
  	trapezoidtrapzz!trapz is an alias for `trapezoid`c                   B    e Zd ZdZddZd Zd Zd Zd Zd Z	d	 Z
d
 Zy)
triang_gena5  A triangular continuous random variable.

    %(before_notes)s

    Notes
    -----
    The triangular distribution can be represented with an up-sloping line from
    ``loc`` to ``(loc + c*scale)`` and then downsloping for ``(loc + c*scale)``
    to ``(loc + scale)``.

    `triang` takes ``c`` as a shape parameter for :math:`0 \le c \le 1`.

    %(after_notes)s

    The standard form is in the range [0, 1] with c the mode.
    The location parameter shifts the start to `loc`.
    The scale parameter changes the width from 1 to `scale`.

    %(example)s

    Nc                 *    |j                  d|d|      S r	  )
triangularr  s       r0   r   ztriang_gen._rvs%%  s    &&q!Q55r2   c                     |dk\  |dk  z  S r	  r   rr  s     r0   r]   ztriang_gen._argcheck(%  s    Q16""r2   c                      t        dddd      gS )Nr  Frn
  ro
  r4
  rd   s    r0   re   ztriang_gen._shape_info+%  s    3x>??r2   c                 `    t        |dk(  ||k  ||k\  |dk7  z  |dk(  gd d d d g||f      }|S )Nr   r   c                     dd| z  z
  S r  r   rc  s     r0   r  z!triang_gen._pdf.<locals>.<lambda>8%  s    a!a%i r2   c                     d| z  |z  S r  r   rc  s     r0   r  z!triang_gen._pdf.<locals>.<lambda>9%      a!eai r2   c                     dd| z
  z  d|z
  z  S r  r   rc  s     r0   r  z!triang_gen._pdf.<locals>.<lambda>:%  s    a1q5kQU&; r2   c                     d| z  S r  r   rc  s     r0   r  z!triang_gen._pdf.<locals>.<lambda>;%  
    a!e r2   ru
  r?   rk   r  r  s       r0   rl   ztriang_gen._pdf.%  sZ     aQq&Q!V,a! 0/;+- A  r2   c                 `    t        |dk(  ||k  ||k\  |dk7  z  |dk(  gd d d d g||f      }|S )Nr   r   c                     d| z  | | z  z
  S r  r   rc  s     r0   r  z!triang_gen._cdf.<locals>.<lambda>D%  s    acAaCi r2   c                     | | z  |z  S rH   r   rc  s     r0   r  z!triang_gen._cdf.<locals>.<lambda>E%  r
  r2   c                 *    | | z  d| z  z
  |z   |dz
  z  S r  r   rc  s     r0   r  z!triang_gen._cdf.<locals>.<lambda>F%  s    qsQqSy1}1&= r2   c                     | | z  S rH   r   rc  s     r0   r  z!triang_gen._cdf.<locals>.<lambda>G%  r
  r2   ru
  r
  s       r0   ro   ztriang_gen._cdf?%  sX    aQq&Q!V,a! 0/=+- A  r2   c           
          t        j                  ||k  t        j                  ||z        dt        j                  d|z
  d|z
  z        z
        S rX   )rJ   rF  r   r  s      r0   rx   ztriang_gen._ppfK%  s?    xxArwwq1u~q!A#!A#1G/GHHr2   c           	          |dz   dz  d|z
  ||z  z   dz  t        j                  d      d|z  dz
  z  |dz   z  |dz
  z  dt        j                  d|z
  ||z  z   d      z  z  dfS )	Nr   r;     rO   r   r:  r  g333333)rJ   r   r  rr  s     r0   r   ztriang_gen._statsN%  sx    3QqsB
AaCE"AaC(!A#.!BHHc!eAaCi#4N2NO 	r2   c                 2    dt        j                  d      z
  S r  r+  rr  s     r0   r   ztriang_gen._entropyT%  s    266!9}r2   r   )r}   r~   r   r   r   r]   re   rl   ro   rx   r   r   r   r2   r0   r
  r
  %  s1    *6#@"
Ir2   r
  triangc                   X     e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Z fd
Zd Z xZS )truncexpon_genad  A truncated exponential continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `truncexpon` is:

    .. math::

        f(x, b) = \frac{\exp(-x)}{1 - \exp(-b)}

    for :math:`0 <= x <= b`.

    `truncexpon` takes ``b`` as a shape parameter for :math:`b`.

    %(after_notes)s

    %(example)s

    c                 @    t        dddt        j                  fd      gS ru  rb   rd   s    r0   re   ztruncexpon_gen._shape_infoq%  r  r2   c                     | j                   |fS rH   r  r	
  s     r0   r   ztruncexpon_gen._get_supportt%      vvqyr2   c                 ^    t        j                  |       t        j                  |        z  S rH   r  rw  s      r0   rl   ztruncexpon_gen._pdfw%  s#    vvqbzBHHaRL=))r2   c                 ^    | t        j                  t        j                  |              z
  S rH   r  rw  s      r0   r   ztruncexpon_gen._logpdf{%  s$    rBFFBHHaRL=)))r2   c                 \    t        j                  |       t        j                  |       z  S rH   r2  rw  s      r0   ro   ztruncexpon_gen._cdf~%  s!    xx|BHHaRL((r2   c                 \    t        j                  |t        j                  |       z         S rH   )rq   r  r  r  s      r0   rx   ztruncexpon_gen._ppf%  s"    288QB<(((r2   c                     t        j                  |       t        j                  |       z
  t        j                  |       z  S rH   r  rw  s      r0   rs   ztruncexpon_gen._sf%  s0    r
RVVQBZ'1"55r2   c                     t        j                  t        j                  |       |t        j                  |       z  z
         S rH   )rJ   r   r   rq   r  r  s      r0   r{   ztruncexpon_gen._isf%  s2    rvvqbzA!$44555r2   c                 2   |dk(  r7d|dz   t        j                  |       z  z
  t        j                  |        z  S |dk(  rFddd||z  d|z  z   dz   z  t        j                  |       z  z
  z  t        j                  |        z  S t        |   ||      S r  )rJ   r   rq   r  r;   r   )r?   r\   r   r  s      r0   r   ztruncexpon_gen._munp%  s     6qsBFFA2J&&"((A2,77!VaQqS1WQYr
223bhhrl]CC 7=A&&r2   c                     t        j                  |      }t        j                  |dz
        d||dz
  z  z   d|z
  z  z   S rz  r	  )r?   r   eBs      r0   r   ztruncexpon_gen._entropy%  s;    VVAYvvbd|Qr1S5z\CF333r2   )r}   r~   r   r   re   r   rl   r   ro   rx   rs   r{   r   r   r  r  s   @r0   r
  r
  [%  s;    *E**))66	'4r2   r
  
truncexponc                 4    t        j                  | |gd      S )Nr   r  )rq   r  log_plog_qs     r0   _log_sumr
  %  s    <<Q//r2   c                 \    t        j                  | |t        j                  dz  z   gd      S )N              ?r   r  )rq   r  rJ   r   r
  s     r0   r 
  r 
  %  s$    <<beeBh/a88r2   c                    t        j                  | |      \  } }|dk  }| dkD  }||z   }d fd}d }t        j                  | t         j                  t         j                        }| |   j
                  r | |   ||         ||<   | |   j
                  r || |   ||         ||<   | |   j
                  r || |   ||         ||<   t        j                  |      S )z3Log of Gaussian probability mass within an intervalr   c                 >    t        t        |      t        |             S rH   )r 
  r   r  s     r0   mass_case_leftz'_log_gauss_mass.<locals>.mass_case_left%  s    a,q/::r2   c                      | |        S rH   r   )r   r   r
  s     r0   mass_case_rightz(_log_gauss_mass.<locals>.mass_case_right%  s    qb1"%%r2   c                 Z    t        j                  t        |        t        |       z
        S rH   )rq   r  r   r  s     r0   mass_case_centralz*_log_gauss_mass.<locals>.mass_case_central%  s$     xx1	1"566r2   )r9  r  )rJ   r  r  r  
complex128r   r  )	r   r   	case_left
case_rightcase_centralr
  r
  r  r
  s	           @r0   _log_gauss_massr
  %  s    q!$DAq QIQJ+,L;&7 ,,qRVV2==
AC|')a	lCI})!J-:GJ-aoqOL773<r2   c                   x     e Zd ZdZd Zd Z fdZd Zd Zd Z	d Z
d	 Zd
 Zd Zd Zd Zd Zd ZddZ xZS )truncnorm_genaw
  A truncated normal continuous random variable.

    %(before_notes)s

    Notes
    -----
    This distribution is the normal distribution centered on ``loc`` (default
    0), with standard deviation ``scale`` (default 1), and truncated at ``a``
    and ``b`` *standard deviations* from ``loc``. For arbitrary ``loc`` and
    ``scale``, ``a`` and ``b`` are *not* the abscissae at which the shifted
    and scaled distribution is truncated.

    .. note::
        If ``a_trunc`` and ``b_trunc`` are the abscissae at which we wish
        to truncate the distribution (as opposed to the number of standard
        deviations from ``loc``), then we can calculate the distribution
        parameters ``a`` and ``b`` as follows::

            a, b = (a_trunc - loc) / scale, (b_trunc - loc) / scale

        This is a common point of confusion. For additional clarification,
        please see the example below.

    %(example)s

    In the examples above, ``loc=0`` and ``scale=1``, so the plot is truncated
    at ``a`` on the left and ``b`` on the right. However, suppose we were to
    produce the same histogram with ``loc = 1`` and ``scale=0.5``.

    >>> loc, scale = 1, 0.5
    >>> rv = truncnorm(a, b, loc=loc, scale=scale)
    >>> x = np.linspace(truncnorm.ppf(0.01, a, b),
    ...                 truncnorm.ppf(0.99, a, b), 100)
    >>> r = rv.rvs(size=1000)

    >>> fig, ax = plt.subplots(1, 1)
    >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
    >>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2)
    >>> ax.set_xlim(a, b)
    >>> ax.legend(loc='best', frameon=False)
    >>> plt.show()

    Note that the distribution is no longer appears to be truncated at
    abscissae ``a`` and ``b``. That is because the *standard* normal
    distribution is first truncated at ``a`` and ``b``, *then* the resulting
    distribution is scaled by ``scale`` and shifted by ``loc``. If we instead
    want the shifted and scaled distribution to be truncated at ``a`` and
    ``b``, we need to transform these values before passing them as the
    distribution parameters.

    >>> a_transformed, b_transformed = (a - loc) / scale, (b - loc) / scale
    >>> rv = truncnorm(a_transformed, b_transformed, loc=loc, scale=scale)
    >>> x = np.linspace(truncnorm.ppf(0.01, a, b),
    ...                 truncnorm.ppf(0.99, a, b), 100)
    >>> r = rv.rvs(size=10000)

    >>> fig, ax = plt.subplots(1, 1)
    >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
    >>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2)
    >>> ax.set_xlim(a-0.1, b+0.1)
    >>> ax.legend(loc='best', frameon=False)
    >>> plt.show()
    c                     ||k  S rH   r   r  s      r0   r]   ztruncnorm_gen._argcheck&  s    1ur2   c                     t        ddt        j                   t        j                  fd      }t        ddt        j                   t        j                  fd      }||gS )Nr   Fra   r   )FTrb   ra  s      r0   re   ztruncnorm_gen._shape_info&  sI    UbffWbff$5}EUbffWbff$5}EBxr2   c                     t        |t              r|j                         }t        |   |t        j                  |      t        j                  |      f      S r	  r	  r  s     r0   r  ztruncnorm_gen._fitstart&  sC    dL)>>#Dw RVVD\266$<,H IIr2   c                 
    ||fS rH   r   r  s      r0   r   ztruncnorm_gen._get_support&  r  r2   c                 N    t        j                  | j                  |||            S rH   r  rn  s       r0   rl   ztruncnorm_gen._pdf!&  r  r2   c                 2    t        |      t        ||      z
  S rH   )r   r
  rn  s       r0   r   ztruncnorm_gen._logpdf$&  s    AA!666r2   c                 N    t        j                  | j                  |||            S rH   r  rn  s       r0   ro   ztruncnorm_gen._cdf'&  r  r2   c           
      T   t        j                  |||      \  }}}t        j                  t        ||      t        ||      z
        }|dkD  }t        j                  |      rFt        j
                  t        j                  | j                  ||   ||   ||                      ||<   |S Ng)rJ   r  r   r
  r  r  r   r   )r?   rk   r   r   logcdfr  s         r0   r   ztruncnorm_gen._logcdf*&  s    %%aA.1aOAq1OAq4IIJTM66!9"&&QqT1Q41)F"G!GHF1Ir2   c                 N    t        j                  | j                  |||            S rH   r-  rn  s       r0   rs   ztruncnorm_gen._sf2&  r.  r2   c           
      T   t        j                  |||      \  }}}t        j                  t        ||      t        ||      z
        }|dkD  }t        j                  |      rFt        j
                  t        j                  | j                  ||   ||   ||                      ||<   |S r
  )rJ   r  r   r
  r  r  r   r   )r?   rk   r   r   logsfr  s         r0   r   ztruncnorm_gen._logsf5&  s    %%aA.1a

?1a0?1a3HHIDL66!9xxQqT1Q41(F!G GHE!Hr2   c                 &   t        |      }t        |      }||z
  }t        j                  t        j                  dt        j                  z  t        j
                  z        |z        }|t        |      z  |t        |      z  z
  d|z  z  }||z   }|S r  )r   rJ   r   r   r   r  r   )	r?   r   r   r!  r"  Zr  Dr  s	            r0   r   ztruncnorm_gen._entropy=&  s}    aLaLEFF2771ruu9rtt+,q011IaL 00QU;Er2   c                    t        j                  |||      \  }}}|dk  }| }d }d }t        j                  |      }||   }	||   }
|	j                  r ||	||   ||         ||<   |
j                  r ||
||   ||         ||<   |S )Nr   c                     t        t        |      t        j                  |       t	        ||      z         }t        j                  |      S rH   )r
  r   rJ   r   r
  rq   	ndtri_exprw   r   r   	log_Phi_xs       r0   ppf_leftz$truncnorm_gen._ppf.<locals>.ppf_leftL&  s9     a!#_Q-B!BDI<<	**r2   c                     t        t        |       t        j                  |        t	        ||      z         }t        j                  |       S rH   )r
  r   rJ   r  r
  rq   r
  r
  s       r0   	ppf_rightz%truncnorm_gen._ppf.<locals>.ppf_rightQ&  sA     qb!1!#1"10E!EGILL+++r2   rJ   r  
empty_liker   )r?   rw   r   r   r
  r
  r
  r
  r  q_leftq_rights              r0   rx   ztruncnorm_gen._ppfF&  s    %%aA.1aE	Z
	+
	,
 mmA9J-;;%fa	lAiLIC	N<<':*NC
O
r2   c                    t        j                  |||      \  }}}|dk  }| }d }d }t        j                  |      }||   }	||   }
|	j                  r ||	||   ||         ||<   |
j                  r ||
||   ||         ||<   |S )Nr   c                     t        t        |      t        j                  |       t	        ||      z         }t        j                  t        j                  |            S rH   )r 
  r   rJ   r   r
  rq   r
  r  r
  s       r0   isf_leftz$truncnorm_gen._isf.<locals>.isf_lefti&  sB    !,q/"$&&)oa.C"CEI<<	 233r2   c                     t        t        |       t        j                  |        t	        ||      z         }t        j                  t        j                  |             S rH   )r 
  r   rJ   r  r
  rq   r
  r  r
  s       r0   	isf_rightz%truncnorm_gen._isf.<locals>.isf_rightn&  sJ    !,r"2"$((A2,A1F"FHILL!3444r2   r
  )r?   rw   r   r   r
  r
  r
  r
  r  r
  r
  s              r0   r{   ztruncnorm_gen._isfb&  s    %%aA.1aE	Z
	4
	5
 mmA9J-;;%fa	lAiLIC	N<<':*NC
O
r2   c                       fd}t        |dk\  ||k(  z  ||k(  z  |||ft        j                  |t        j                  g      t        j                        S )Nc                 0  	 
j                  t        j                  ||g      ||      \  }}|| g}ddg}t        d| dz         D ]J  	t	        ||||gg	fdd      }t        j
                  |      	dz
  |d   z  z   }|j                  |       L |d   S )z
            Returns n-th moment. Defined only if n >= 0.
            Function cannot broadcast due to the loop over n
            r   r   c                     | |dz
  z  z  S rX   r   )rk   r^  r
  s     r0   r  z:truncnorm_gen._munp.<locals>.n_th_moment.<locals>.<lambda>&  s    q1qs8| r2   r  r	  r  )rl   rJ   r   r  r   r  r  )r\   r   r   pApBprobsr  r  mkr
  r?   s            @r0   n_th_momentz(truncnorm_gen._munp.<locals>.n_th_moment&  s    
 YYrzz1a&11a8FB"IE!fG1ac] #
 "%%!Q";qJVVD\QqSGBK$77r"# 2;r2   r   rH  r  )r?   r\   r   r   r
  s   `    r0   r   ztruncnorm_gen._munp&  sR    	& 16a1f-a81a),,{BJJ<H&&" 	"r2   c                     | j                  t        j                  ||g      ||      \  }}d }t        j                  |d      } ||||||      S )Nc                 H   ||z
  }|}|| g}t        ||| |ggd d      }dt        j                  |      z   }	t        ||| |z
  ||z
  ggd d      }dt        j                  |      z   }
t        ||| |ggd d      }d|z  t        j                  |      z   }t        ||| |ggd d      }d	|	z  t        j                  |      z   }||d
|	z  d|dz  z  z   z  z   }|t        j                  |
d      z  }||d|z  d	|z  d|	z  |dz  z
  z  z   z  z   }||
dz  z  d	z
  }||
||fS )Nc                     | |z  S rH   r   r]  s     r0   r  zGtruncnorm_gen._stats.<locals>._truncnorm_stats_scalar.<locals>.<lambda>&  s
    1Q3 r2   r   r  r   c                     | |z  S rH   r   r]  s     r0   r  zGtruncnorm_gen._stats.<locals>._truncnorm_stats_scalar.<locals>.<lambda>&  s
    1 r2   c                     | |dz  z  S r  r   r]  s     r0   r  zGtruncnorm_gen._stats.<locals>._truncnorm_stats_scalar.<locals>.<lambda>&      1QT6 r2   rO   c                     | |dz  z  S r+
  r   r]  s     r0   r  zGtruncnorm_gen._stats.<locals>._truncnorm_stats_scalar.<locals>.<lambda>&  r
  r2   r  r  r  r  )r   rJ   r  r  )r   r   r
  r
  r  r  r<  r
  r  r  r=  r  m4mu3r>  mu4r?  s                    r0   _truncnorm_stats_scalarz5truncnorm_gen._stats.<locals>._truncnorm_stats_scalar&  sg   bBB"IEeeaV_6F()+DRVVD\!BeeadAbD\%:<L()+D bffTl"CeeaV_6I()+D2t$BeeaV_6I()+D2t$BrRUQr1uW_--CrxxS))Br2b51R42A#6677CsAv!BsB?"r2   )r  )excluded)pdfrJ   rQ  rB  )r?   r   r   r  r
  r
  r
  _truncnorm_statss           r0   r   ztruncnorm_gen._stats&  sT    "((Aq6*Aq1B	#4 <<(?1=?1b"g66r2   r  )r}   r~   r   r   r]   re   r  r   rl   r   ro   r   rs   r   r   rx   r{   r   r   r  r  s   @r0   r
  r
  %  sU    >@
J-7-,8:"07r2   r
  	truncnorm)r   r   c                        e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zd Zd Zd Zd Zd Ze ee       fd              Z xZS )truncpareto_genac  An upper truncated Pareto continuous random variable.

    %(before_notes)s

    See Also
    --------
    pareto : Pareto distribution

    Notes
    -----
    The probability density function for `truncpareto` is:

    .. math::

        f(x, b, c) = \frac{b}{1 - c^{-b}} \frac{1}{x^{b+1}}

    for :math:`b > 0`, :math:`c > 1` and :math:`1 \le x \le c`.

    `truncpareto` takes `b` and `c` as shape parameters for :math:`b` and
    :math:`c`.

    Notice that the upper truncation value :math:`c` is defined in
    standardized form so that random values of an unscaled, unshifted variable
    are within the range ``[1, c]``.
    If ``u_r`` is the upper bound to a scaled and/or shifted variable,
    then ``c = (u_r - loc) / scale``. In other words, the support of the
    distribution becomes ``(scale + loc) <= x <= (c*scale + loc)`` when
    `scale` and/or `loc` are provided.

    %(after_notes)s

    References
    ----------
    .. [1] Burroughs, S. M., and Tebbens S. F.
        "Upper-truncated power laws in natural systems."
        Pure and Applied Geophysics 158.4 (2001): 741-757.

    %(example)s

    c                     t        dddt        j                  fd      }t        dddt        j                  fd      }||gS )Nr   Fr   r  r  r   rb   )r?   rc  r  s      r0   re   ztruncpareto_gen._shape_info&  s;    US"&&M>BUS"&&M>BBxr2   c                     |dkD  |dkD  z  S Nr   r   r   r?   r   r  s      r0   r]   ztruncpareto_gen._argcheck&  s    B1r6""r2   c                     | j                   |fS rH   r  r  s      r0   r   ztruncpareto_gen._get_support&  r
  r2   c                 2    |||dz    z  z  dd||z  z  z
  z  S rX   r   r?   rk   r   r  s       r0   rl   ztruncpareto_gen._pdf&  s'    1!f9}AadF
++r2   c           	          t        j                  |      t        j                  t        j                  | t        j                  |      z               z
  |dz   t        j                  |      z  z
  S rX   )rJ   r   r  r  s       r0   r   ztruncpareto_gen._logpdf&  sM    vvay266288QBrvvayL#9"9::ac266!9_LLr2   c                 ,    d|| z  z
  dd||z  z  z
  z  S rX   r   r  s       r0   ro   ztruncpareto_gen._cdf&  s#    ArE	a!AqD&j))r2   c                 n    t        j                  || z         t        j                  d||z  z        z
  S r  r0  r  s       r0   r   ztruncpareto_gen._logcdf&  s/    xxQB"((2ad7"333r2   c                 >    t        ddd||z  z  z
  |z  z
  d|z        S Nr   r  r  r?   rw   r   r  s       r0   rx   ztruncpareto_gen._ppf&  s(    1AadF
A~%r!t,,r2   c                 8    || z  d||z  z  z
  dd||z  z  z
  z  S rX   r   r  s       r0   rs   ztruncpareto_gen._sf '  s+    A2!Q$1qAv:..r2   c                 ~    t        j                  || z  d||z  z  z
        t        j                  d||z  z        z
  S r
  r  r  s       r0   r   ztruncpareto_gen._logsf'  s9    vva!ea1fn%AqD(999r2   c                 J    t        d||z  z  dd||z  z  z
  |z  z   d|z        S r
  r  r  s       r0   r{   ztruncpareto_gen._isf'  s0    1QT6Q1a4ZN*BqD11r2   c                     t        j                  |dd||z  z  z
  z        |dz   t        j                  |      ||z  dz
  z  d|z  z
  z  z    S rX   r+  r  s      r0   r   ztruncpareto_gen._entropy	'  sW    1qAv:'aC"&&)QTAX.1456 7 	7r2   c                     ||k(  j                         r$|t        j                  |      z  dd||z  z  z
  z  S |||z
  z  ||z  ||z  z
  z  ||z  dz
  z  S rX   )r   rJ   r   )r?   r\   r   r  s       r0   r   ztruncpareto_gen._munp'  s^    F<<>RVVAY;!a1f*--!91q!t,1q99r2   c                     t        |t              r|j                         }t        j	                  |      \  }}}t        |      |z
  |z  }||||fS rH   )r9   r%   r  rP	  r=   rY  )r?   r@   r   r)   r*   r  s         r0   r  ztruncpareto_gen._fitstart'  sM    dL)>>#D

4(3Y_e#!S%r2   c                     !"#$% |j                  dd      rt        &   g|i |S d #d ""#fd%fd $%fd}$fd!d !"fd		}d
 }& fd}t         ||      }|\  }	}
}}j	                         j                         c$%t        j                  $t        j                         }|	|
||t        d      |
|||	 !"fd}t        j                  $t        j                         }|}d}|dz
  }|t        j                   kD  r[ ||       ||      z  dk\  rG|dz  }|t        j                  d|      z
  }|t        j                   kD  r ||       ||      z  dk\  rG|t        j                   kD  s |g|i |S t        |||f      }|j                  s |g|i |S |j                  dz
  }|dz
  }d}|t        j                   kD  r[ ||       ||      z  dk\  rG|dz  }|t        j                  d|      z
  }|t        j                   kD  r ||       ||      z  dk\  rG|t        j                   kD  s |g|i |S t        |||f      }|j                  s |g|i |S |j                  } !|      }  ||      } |||      }|z
  |z  }t	        d #|      z  d "|      dz
  z        }||k  s |g|i |S |}|dz
  }d}|t        j                   kD  rN |||	       |||	      z  dk\  r8|dz  }|d|z  z
  }|t        j                   kD  r |||	       |||	      z  dk\  r8|t        j                   kD  s |g|i |S t        ||	f||f      }|j                  s |g|i |S |j                  } !|      }  ||      }|	}n||n ||
|      }|xs  !|      }|
xs	   ||      }|$j	                         |z
  dk  rt        dd|      |
r2|0|r.j                         |
|z  |z   kD  rt        dd  ||            |	|z
  |z  } #|      }t        j                  |      }d|z  |k  s |g|i |S d|z  d||z
  z  z   }t        j                  d|z  d      }	 t        |||f||f      }|j                  s |g|i |S |j                  }n|	}||z   $k  sF|r&t        j                  |t        j                         }n !|      }t        j                  |d      }||z  |z   %kD  s-  ||      }t        j                  |t        j                        }t        j                    j#                  ||            r|dkD  s |g|i |S ||||f}|9|7 |g|i |} j%                  |      } j%                  |      }||k  r|S |S # t        $ r |}Y w xY w)Nr  Fc                 R    t        j                  t        j                  |             S rH   )rJ   r   r   r   s    r0   log_meanz%truncpareto_gen.fit.<locals>.log_mean '  s    77266!9%%r2   c                 8    dt        j                  d| z        z  S rX   )rJ   r   r   s    r0   	harm_meanz&truncpareto_gen.fit.<locals>.harm_mean#'  s    RWWQqS\>!r2   c                     |z
  |z  } |      } 	|      }|dz
  |z  }d|dz
  |dd| z  z
  |z  t        j                  |       z  z
  z  z
  |z  S rX   r+  )
r  r)   r*   r  harm_mlog_mquotr@   r  r  s
          r0   get_bz"truncpareto_gen.fit.<locals>.get_b&'  si    c5 Aq\FQKE1He#DaDA!GV+;BFF1I+E$EFFMMr2   c                     | z
  |z  S rH   r   )r)   r*   mxs     r0   get_cz"truncpareto_gen.fit.<locals>.get_c-'  s    He##r2   c                 <    |r|z
  }|S | r| z  z
  | dz
  z  }|S y rX   r   )r  r   r)   rR  r  s      r0   get_locz$truncpareto_gen.fit.<locals>.get_loc0'  s7    6k
"urzBF+
 r2   c                     | z
  S rH   r   )r)   rR  s    r0   r	  z&truncpareto_gen.fit.<locals>.get_scale8'  s    8Or2   c                      	|       } | |      }|
 || |      n|} 
| z
  |z        }dd|dz
  ||dz   z  |z
  z  z   dd|dz   z  z
  z  |z  z
  S rX   r   )r)   r  r*   r  r   r  r@   r  r  r	  r  s         r0   r  z$truncpareto_gen.fit.<locals>.dL_dLoc>'  s|     cNEc5!A(*
ae$As
E12FQUQ1X\22q1ac7{CfLLLr2   c                 P    | t        j                  | |z  d| |z  z
  z        |z  z
  S rX   r0  )r   logclogms      r0   dL_dBz"truncpareto_gen.fit.<locals>.dL_dBG'  s.     rxx$!af* 56===r2   c                 2    t        t        
  | g|i |S rH   )r;   r
  r=   )r@   rA   kwargsr  r?   s      r0   fallbackz%truncpareto_gen.fit.<locals>.fallbackM'  s    $3DJ4J6JJr2   z2All parameters fixed.There is nothing to optimize.c                      |       } | |      } | z
  |z        }dd|dz
  z  z   t        j                  |      z  |z  dz
  S rX   r+  )r)   r*   r  r  r@   r  r	  r  s       r0   cond_bz#truncpareto_gen.fit.<locals>.cond_b^'  sT    %cNEc5)A&s
E'9:F1Q3K266!94v=AAr2   r   r   r   r  gMbP?rO   truncparetor  rH   )r-   r;   r=   r  r@  rY  rJ   r  rc   r   r  r&   r  r  rD  r   r   r]   r  )'r?   r@   rA   r/   r   r  r&  r)  r  r  r  r   r   mn_infr+  rM   r  rL   r  r)   r*   r  r   std_data
up_bound_br%  r$  params_overrideparams_supernllf_override
nllf_superr  r  r	  r  r  rR  r  r  s'   ``                             @@@@@@@r0   r=   ztruncpareto_gen.fit'  s    88J&7;t3d3d33	&	"	N	$			M 	M	>	K 1tT4H
%/"b"dFTXXZBb266'*NN$& = > >ZDLV^zB b266'2!"&&("6N6&>9Q>FA#bhhr1o5F "&&("6N6&>9Q> '#D848488!&662BC}}#D848488 D!"&&(#FOGFO;q@FA#bhhr1o5F "&&(#FOGFO;q@ '#D848488!'FF3CD}}#D848488hh!##u%!S%( 3J- 8H#5!5!"Ih$7$9!:<
J#D848488  !'#FB/%fb12567FA#ad]F	 '#FB/%fb12567 '#D848488!'B5+16*:<}}#D848488hh!##u% *$F0CC,inE'eC'A DHHJ$5$9"=CC t'V88:6	D 00&}A-23->@ @ z 3J-)vvay$#D8484884!TD[/1afa0%edD\/5v.>@C =='<t<t<<A  c	Rll30!#UA.%r!c5!AQ'At~~a+,%!)D040400QU*<FN
 $D84848L IIot<M<6JM)##E " As   -)W% W% %W43W4)r}   r~   r   r   re   r]   r   rl   r   ro   r   rx   rs   r   r{   r   r   r  rE   r
   r   r=   r  r  s   @r0   r
  r
  &  ss    'R
#,M*4-/:27:  M*Z + Zr2   r
  r,  c                   :    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
y	)
tukeylambda_gena*  A Tukey-Lamdba continuous random variable.

    %(before_notes)s

    Notes
    -----
    A flexible distribution, able to represent and interpolate between the
    following distributions:

    - Cauchy                (:math:`lambda = -1`)
    - logistic              (:math:`lambda = 0`)
    - approx Normal         (:math:`lambda = 0.14`)
    - uniform from -1 to 1  (:math:`lambda = 1`)

    `tukeylambda` takes a real number :math:`lambda` (denoted ``lam``
    in the implementation) as a shape parameter.

    %(after_notes)s

    %(example)s

    c                 ,    t        j                  |      S rH   rY  r?   lams     r0   r]   ztukeylambda_gen._argcheck(  s    {{3r2   c                 ^    t        ddt        j                   t        j                  fd      gS )Nr8  Fr  rb   rd   s    r0   re   ztukeylambda_gen._shape_info(  s%    5%266'266):NKLLr2   c                 P   t        j                  t        j                  ||            }||dz
  z  t        j                  d|z
        |dz
  z  z   }dt        j                  |      z  }t        j                  |dk  t        |      dt        j                  |      z  k  z  |d      S )Nr   r   r   r   )rJ   r   rq   tklmbdarF  r  )r?   rk   r8  Fxr  s        r0   rl   ztukeylambda_gen._pdf(  s    ZZ

1c*+#c']bjj2.#c'::Bxxc!fs2::c?/B&BCRMMr2   c                 .    t        j                  ||      S rH   )rq   r;  )r?   rk   r8  s      r0   ro   ztukeylambda_gen._cdf(  s    zz!S!!r2   c                 ^    t        j                  ||      t        j                  | |      z
  S rH   )rq   rm  rk  )r?   rw   r8  s      r0   rx   ztukeylambda_gen._ppf"(  s%    yyC 2;;r3#777r2   c                 2    dt        |      dt        |      fS r  )_tlvar_tlkurtr7  s     r0   r   ztukeylambda_gen._stats%(  s    &+q'#,..r2   c                 B    fd}t        j                  |dd      d   S )Nc                 n    t        j                  t        | dz
        t        d| z
  dz
        z         S rX   )rJ   r   r  )r  r8  s    r0   integz'tukeylambda_gen._entropy.<locals>.integ)(  s/    66#aQ-AaCQ788r2   r   r   )r   rW  )r?   r8  rD  s    ` r0   r   ztukeylambda_gen._entropy((  s     	9~~eQ*1--r2   N)r}   r~   r   r   r]   re   rl   ro   rx   r   r   r   r2   r0   r5  r5  '  s,    , MN"8/.r2   r5  tukeylambdac                       e Zd Zd Zy)FitUniformFixedScaleDataErrorc                      d| d| d| _         y )NzInvalid values in `data`.  Maximum likelihood estimation with the uniform distribution and fixed scale requires that np.ptp(data) <= fscale, but np.ptp(data) = z and fscale = r  rF  )r?   r	  r   s      r0   rI  z&FitUniformFixedScaleDataError.__init__2(  s$    ::= ?xq" 		r2   N)r}   r~   r   rI  r   r2   r0   rG  rG  1(  s    
r2   rG  c                   L    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
ed
        Zy)uniform_gena  A uniform continuous random variable.

    In the standard form, the distribution is uniform on ``[0, 1]``. Using
    the parameters ``loc`` and ``scale``, one obtains the uniform distribution
    on ``[loc, loc + scale]``.

    %(before_notes)s

    %(example)s

    c                     g S rH   r   rd   s    r0   re   zuniform_gen._shape_infoG(  r   r2   Nc                 (    |j                  dd|      S r  )r  r   s      r0   r   zuniform_gen._rvsJ(  s    ##Cd33r2   c                     d||k(  z  S r  r   r   s     r0   rl   zuniform_gen._pdfM(  s    AF|r2   c                     |S rH   r   r   s     r0   ro   zuniform_gen._cdfP(      r2   c                     |S rH   r   r   s     r0   rx   zuniform_gen._ppfS(  rO  r2   c                      y)N)r   gUUUUUU?r   g333333r   rd   s    r0   r   zuniform_gen._statsV(  s    #r2   c                      yr  r   rd   s    r0   r   zuniform_gen._entropyY(  r<  r2   c                    t        |      dkD  rt        d      |j                  dd      }|j                  dd      }t        |       ||t	        d      t        j                  |      }t        j                  |      j                         st	        d      |a|&|j                         }t        j                  |      }n{|}|j                         |z
  }|j                         |k  rSt        d|||z   	      t        j                  |      }||kD  rt        ||
      |j                         d||z
  z  z
  }|}t        |      t        |      fS )a	  
        Maximum likelihood estimate for the location and scale parameters.

        `uniform.fit` uses only the following parameters.  Because exact
        formulas are used, the parameters related to optimization that are
        available in the `fit` method of other distributions are ignored
        here.  The only positional argument accepted is `data`.

        Parameters
        ----------
        data : array_like
            Data to use in calculating the maximum likelihood estimate.
        floc : float, optional
            Hold the location parameter fixed to the specified value.
        fscale : float, optional
            Hold the scale parameter fixed to the specified value.

        Returns
        -------
        loc, scale : float
            Maximum likelihood estimates for the location and scale.

        Notes
        -----
        An error is raised if `floc` is given and any values in `data` are
        less than `floc`, or if `fscale` is given and `fscale` is less
        than ``data.max() - data.min()``.  An error is also raised if both
        `floc` and `fscale` are given.

        Examples
        --------
        >>> import numpy as np
        >>> from scipy.stats import uniform

        We'll fit the uniform distribution to `x`:

        >>> x = np.array([2, 2.5, 3.1, 9.5, 13.0])

        For a uniform distribution MLE, the location is the minimum of the
        data, and the scale is the maximum minus the minimum.

        >>> loc, scale = uniform.fit(x)
        >>> loc
        2.0
        >>> scale
        11.0

        If we know the data comes from a uniform distribution where the support
        starts at 0, we can use `floc=0`:

        >>> loc, scale = uniform.fit(x, floc=0)
        >>> loc
        0.0
        >>> scale
        13.0

        Alternatively, if we know the length of the support is 12, we can use
        `fscale=12`:

        >>> loc, scale = uniform.fit(x, fscale=12)
        >>> loc
        1.5
        >>> scale
        12.0

        In that last example, the support interval is [1.5, 13.5].  This
        solution is not unique.  For example, the distribution with ``loc=2``
        and ``scale=12`` has the same likelihood as the one above.  When
        `fscale` is given and it is larger than ``data.max() - data.min()``,
        the parameters returned by the `fit` method center the support over
        the interval ``[data.min(), data.max()]``.

        r   r>  r   Nr   r   r   r  r  )r	  r   r   )r  r.   r-   r1   r   rJ   r   r   r   r@  r	  rY  rD  rG  rA  )	r?   r@   rA   r/   r   r   r)   r*   r	  s	            r0   r=   zuniform_gen.fit\(  sF   V t9q=122xx%(D)$T* 2 ) * * zz${{4 $$&CDD> >|hhjt 
S(88:#&y3;OO &&,CV|3FKK ((*sFSL11CE Sz5<''r2   r   )r}   r~   r   r   re   r   rl   ro   rx   r   r   rE   r=   r   r2   r0   rJ  rJ  ;(  s@    
4$ R( R(r2   rJ  r  c                        e Zd ZdZd Zd ZddZ ee       fd       Z	d Z
d Zd Zd	 Zd
 Z eed      	 	 d fd	       Ze eed       fd              Z xZS )vonmises_genaU  A Von Mises continuous random variable.

    %(before_notes)s

    See Also
    --------
    scipy.stats.vonmises_fisher : Von-Mises Fisher distribution on a
                                  hypersphere

    Notes
    -----
    The probability density function for `vonmises` and `vonmises_line` is:

    .. math::

        f(x, \kappa) = \frac{ \exp(\kappa \cos(x)) }{ 2 \pi I_0(\kappa) }

    for :math:`-\pi \le x \le \pi`, :math:`\kappa \ge 0`. :math:`I_0` is the
    modified Bessel function of order zero (`scipy.special.i0`).

    `vonmises` is a circular distribution which does not restrict the
    distribution to a fixed interval. Currently, there is no circular
    distribution framework in SciPy. The ``cdf`` is implemented such that
    ``cdf(x + 2*np.pi) == cdf(x) + 1``.

    `vonmises_line` is the same distribution, defined on :math:`[-\pi, \pi]`
    on the real line. This is a regular (i.e. non-circular) distribution.

    Note about distribution parameters: `vonmises` and `vonmises_line` take
    ``kappa`` as a shape parameter (concentration) and ``loc`` as the location
    (circular mean). A ``scale`` parameter is accepted but does not have any
    effect.

    Examples
    --------
    Import the necessary modules.

    >>> import numpy as np
    >>> import matplotlib.pyplot as plt
    >>> from scipy.stats import vonmises

    Define distribution parameters.

    >>> loc = 0.5 * np.pi  # circular mean
    >>> kappa = 1  # concentration

    Compute the probability density at ``x=0`` via the ``pdf`` method.

    >>> vonmises.pdf(0, loc=loc, kappa=kappa)
    0.12570826359722018

    Verify that the percentile function ``ppf`` inverts the cumulative
    distribution function ``cdf`` up to floating point accuracy.

    >>> x = 1
    >>> cdf_value = vonmises.cdf(x, loc=loc, kappa=kappa)
    >>> ppf_value = vonmises.ppf(cdf_value, loc=loc, kappa=kappa)
    >>> x, cdf_value, ppf_value
    (1, 0.31489339900904967, 1.0000000000000004)

    Draw 1000 random variates by calling the ``rvs`` method.

    >>> sample_size = 1000
    >>> sample = vonmises(loc=loc, kappa=kappa).rvs(sample_size)

    Plot the von Mises density on a Cartesian and polar grid to emphasize
    that it is a circular distribution.

    >>> fig = plt.figure(figsize=(12, 6))
    >>> left = plt.subplot(121)
    >>> right = plt.subplot(122, projection='polar')
    >>> x = np.linspace(-np.pi, np.pi, 500)
    >>> vonmises_pdf = vonmises.pdf(x, loc=loc, kappa=kappa)
    >>> ticks = [0, 0.15, 0.3]

    The left image contains the Cartesian plot.

    >>> left.plot(x, vonmises_pdf)
    >>> left.set_yticks(ticks)
    >>> number_of_bins = int(np.sqrt(sample_size))
    >>> left.hist(sample, density=True, bins=number_of_bins)
    >>> left.set_title("Cartesian plot")
    >>> left.set_xlim(-np.pi, np.pi)
    >>> left.grid(True)

    The right image contains the polar plot.

    >>> right.plot(x, vonmises_pdf, label="PDF")
    >>> right.set_yticks(ticks)
    >>> right.hist(sample, density=True, bins=number_of_bins,
    ...            label="Histogram")
    >>> right.set_title("Polar plot")
    >>> right.legend(bbox_to_anchor=(0.15, 1.06))

    c                 @    t        dddt        j                  fd      gS )NrD  Fr   ra   rb   rd   s    r0   re   zvonmises_gen._shape_infoU)  s    7EArvv;FGGr2   c                     |dk\  S r  r   rT  s     r0   r]   zvonmises_gen._argcheckX)  s    zr2   c                 *    |j                  d||      S )Nr   r  )vonmises)r?   rD  r   r   s       r0   r   zvonmises_gen._rvs[)  s    $$S%d$;;r2   c                     t        |   |i |}t        j                  |t        j                  z   dt        j                  z        t        j                  z
  S r  )r;   r  rJ   modr   )r?   rA   r/   r  r  s       r0   r  zvonmises_gen.rvs^)  s@    gk4(4(vvcBEEk1RUU7+bee33r2   c                     t        j                  |t        j                  |      z        dt         j                  z  t        j
                  |      z  z  S r  )rJ   r   rq   cosm1r   r
  rF  s      r0   rl   zvonmises_gen._pdfc)  s:    
 vveBHHQK'(AbeeGBFF5M,ABBr2   c                     |t        j                  |      z  t        j                  dt        j                  z        z
  t        j                  t        j
                  |            z
  S r  )rq   r]  rJ   r   r   r
  rF  s      r0   r   zvonmises_gen._logpdfj)  s@    rxx{"RVVAbeeG_4rvvbffUm7LLLr2   c                 .    t        j                  ||      S rH   )r   von_mises_cdfrF  s      r0   ro   zvonmises_gen._cdfn)  s    ##E1--r2   c                      yr$  r   rT  s     r0   _stats_skipzvonmises_gen._stats_skipq)  r%  r2   c                     | t        j                  |      z  t        j                  |      z  t        j                  dt        j
                  z  t        j                  |      z        z   |z   S r  )rq   i1er
  rJ   r   r   rT  s     r0   r   zvonmises_gen._entropyt)  sV     &6q255y266%=01249: 	;r2   z        The default limits of integration are endpoints of the interval
        of width ``2*pi`` centered at `loc` (e.g. ``[-pi, pi]`` when
        ``loc=0``).

r   c           	          t         j                   t         j                  }
}	|||	z   }|||
z   }t        |   |||||||fi |S rH   )rJ   r   r;   expect)r?   rW  rA   r)   r*   lbubconditionalr/   r  r  r  s              r0   rf  zvonmises_gen.expect)  s_     %%B:rB:rBw~dD##R[B<@B 	Br2   a          Fit data is assumed to represent angles and will be wrapped onto the
        unit circle. `f0` and `fscale` are ignored; the returned shape is
        always the maximum likelihood estimate and the scale is always
        1. Initial guesses are ignored.

c                    |j                  dd      rt        |   |g|i |S t        | |||      \  }}}}| j                  t
        j                   k(  rt        |   |g|i |S t        j                  |dt
        j                  z        }d }d }||n ||      }	||n |||	      }
t        j                  |	t
        j                  z   dt
        j                  z        t
        j                  z
  }	|
|	dfS )Nr  FrO   c                 ,    t        j                  |       S rH   )r  circmean)r@   s    r0   find_muz!vonmises_gen.fit.<locals>.find_mu)  s    >>$''r2   c                 j   t        j                  t        j                  || z
              t        |       z  dk(  rydkD  rNfd}dz
  z  dz   z  }d|z  } ||      dk\  r|S  ||      dk  r|S t	        |d||f      }|j
                  S t        j                  t              j                  S )Nr   g 7yACr   c                 `    t        j                  |       t        j                  |       z  z
  S rH   )rq   rd  r
  )rD  r  s    r0   solve_for_kappaz=vonmises_gen.fit.<locals>.find_kappa.<locals>.solve_for_kappa)  s#    66%=6::r2   rO   r  )r,   r  )	rJ   r  r  r  r&   r  r  rA  r  )r@   r)   rp  lower_boundupper_boundroot_resr  s         @r0   
find_kappaz$vonmises_gen.fit.<locals>.find_kappa)  s     rvvcDj)*3t94A Av Q;  1gqsmm #;/14&&$[1Q6&&*?84?3M OH#==( xx+++r2   r   )r-   r;   r=   r  r   rJ   r   r[  )r?   r@   rA   r/   r  r   r   rm  rt  r)   rn  r  s              r0   r=   zvonmises_gen.fit)  s     88J&7;t3d3d33%@tAEt&M"fdF66beeV7;t3d3d33 vvdAI&	(6	,r &dGDM ,*T32GffS255[!bee),ruu4c1}r2   r   )Nr   r   r   NNF)r}   r~   r   r   re   r]   r   r
   r   r  rl   r   ro   rb  r   r   rf  rE   r=   r  r  s   @r0   rU  rU  (  s    ^~H< M*4 +4CM. 
; } 5  FJ 
B	
B } 5/ 0
N0 Nr2   rU  rY  vonmises_linec                   r    e Zd ZdZej
                  Zd ZddZd Z	d Z
d Zd Zd	 Zd
 Zd Zd Zd Zd Zy)rf  aX  A Wald continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `wald` is:

    .. math::

        f(x) = \frac{1}{\sqrt{2\pi x^3}} \exp(- \frac{ (x-1)^2 }{ 2x })

    for :math:`x >= 0`.

    `wald` is a special case of `invgauss` with ``mu=1``.

    %(after_notes)s

    %(example)s
    c                     g S rH   r   rd   s    r0   re   zwald_gen._shape_info*  r   r2   Nc                 *    |j                  dd|      S rJ  rK  r   s      r0   r   zwald_gen._rvs*  s      c 55r2   c                 .    t         j                  |d      S r  )re  rl   r   s     r0   rl   zwald_gen._pdf*  s    }}Q$$r2   c                 .    t         j                  |d      S r  )re  ro   r   s     r0   ro   zwald_gen._cdf*      }}Q$$r2   c                 .    t         j                  |d      S r  )re  rs   r   s     r0   rs   zwald_gen._sf*  s    ||As##r2   c                 .    t         j                  |d      S r  )re  rx   r   s     r0   rx   zwald_gen._ppf*  r{  r2   c                 .    t         j                  |d      S r  )re  r{   r   s     r0   r{   zwald_gen._isf*  r{  r2   c                 .    t         j                  |d      S r  )re  r   r   s     r0   r   zwald_gen._logpdf*      3''r2   c                 .    t         j                  |d      S r  )re  r   r   s     r0   r   zwald_gen._logcdf*  r  r2   c                 .    t         j                  |d      S r  )re  r   r   s     r0   r   zwald_gen._logsf*  s    q#&&r2   c                      y)N)r   r   r;  rN  r   rd   s    r0   r   zwald_gen._stats!*  s    "r2   c                 ,    t         j                  d      S r  )re  r   rd   s    r0   r   zwald_gen._entropy$*  s      %%r2   r   )r}   r~   r   r   r   r  r  re   r   rl   ro   rs   rx   r{   r   r   r   r   r   r   r2   r0   rf  rf  )  sP    ( "44M6%%$%%(('#&r2   rf  rL  c                   :    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
y	)
wrapcauchy_gena  A wrapped Cauchy continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `wrapcauchy` is:

    .. math::

        f(x, c) = \frac{1-c^2}{2\pi (1+c^2 - 2c \cos(x))}

    for :math:`0 \le x \le 2\pi`, :math:`0 < c < 1`.

    `wrapcauchy` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                     |dkD  |dk  z  S r	  r   rr  s     r0   r]   zwrapcauchy_gen._argcheckA*  r	  r2   c                      t        dddd      gS )Nr  F)r   r   r  r4
  rd   s    r0   re   zwrapcauchy_gen._shape_infoD*  s    3v~>??r2   c                     d||z  z
  dt         j                  z  d||z  z   d|z  t        j                  |      z  z
  z  z  S r  r  r  s      r0   rl   zwrapcauchy_gen._pdfG*  s?    AaC!BEE'1QqS51RVVAY#6788r2   c                 h    d }d }d|z   d|z
  z  }t        |t        j                  k  ||f||      S )Nc                     dt         j                  z  t        j                  |t        j                  | dz        z        z  S r*  rJ   r   r}  r  rk   crs     r0   r  zwrapcauchy_gen._cdf.<locals>.f1M*  s.    RUU7RYYr"&&1+~666r2   c           	          ddt         j                  z  t        j                  |t        j                  dt         j                  z  | z
  dz        z        z  z
  S r*  r  r  s     r0   r  zwrapcauchy_gen._cdf.<locals>.f2Q*  sA    qw2bffagk1_.E+E!FFFFr2   r   r&  )r   rJ   r   )r?   rk   r  r  r  r  s         r0   ro   zwrapcauchy_gen._cdfK*  s=    	7	G !ea!e_!bee)aWr::r2   c           
      v   d|z
  d|z   z  }dt        j                  |t        j                  t         j                  |z        z        z  }dt         j                  z  dt        j                  |t        j                  t         j                  d|z
  z        z        z  z
  }t        j                  |dk  ||      S )Nr   rO   r   r   )rJ   r}  r  r   rF  )r?   rw   r  r7  rcqrcmqs         r0   rx   zwrapcauchy_gen._ppfX*  s    1us1uo		#bffRUU1Wo-..wq3rvvbeeQqSk':#:;;;xxE	3--r2   c                 `    t        j                  dt         j                  z  d||z  z
  z        S r  r   rr  s     r0   r   zwrapcauchy_gen._entropy^*  s%    vvagq1uo&&r2   c                     t        |t              r|j                         }dt        j                  |      t        j
                  |      dt        j                  z  z  fS r  )r9   r%   r  rJ   r@  r	  r   )r?   r@   s     r0   r  zwrapcauchy_gen._fitstarta*  sD     dL)>>#DBFF4L"&&,"%%"888r2   N)r}   r~   r   r   r]   re   rl   ro   rx   r   r  r   r2   r0   r  r  +*  s+    *!@9;.'9r2   r  
wrapcauchyc                   N    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 ZddZy)gennorm_gena0  A generalized normal continuous random variable.

    %(before_notes)s

    See Also
    --------
    laplace : Laplace distribution
    norm : normal distribution

    Notes
    -----
    The probability density function for `gennorm` is [1]_:

    .. math::

        f(x, \beta) = \frac{\beta}{2 \Gamma(1/\beta)} \exp(-|x|^\beta),

    where :math:`x` is a real number, :math:`\beta > 0` and
    :math:`\Gamma` is the gamma function (`scipy.special.gamma`).

    `gennorm` takes ``beta`` as a shape parameter for :math:`\beta`.
    For :math:`\beta = 1`, it is identical to a Laplace distribution.
    For :math:`\beta = 2`, it is identical to a normal distribution
    (with ``scale=1/sqrt(2)``).

    References
    ----------

    .. [1] "Generalized normal distribution, Version 1",
           https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1

    .. [2] Nardon, Martina, and Paolo Pianca. "Simulation techniques for
           generalized Gaussian densities." Journal of Statistical
           Computation and Simulation 79.11 (2009): 1317-1329

    .. [3] Wicklin, Rick. "Simulate data from a generalized Gaussian
           distribution" in The DO Loop blog, September 21, 2016,
           https://blogs.sas.com/content/iml/2016/09/21/simulate-generalized-gaussian-sas.html

    %(example)s

    c                 @    t        dddt        j                  fd      gS Nrg  Fr   r  rb   rd   s    r0   re   zgennorm_gen._shape_info*      651bff+~FGGr2   c                 L    t        j                  | j                  ||            S rH   r  r?   rk   rg  s      r0   rl   zgennorm_gen._pdf*  s    vvdll1d+,,r2   c                     t        j                  d|z        t        j                  d|z        z
  t	        |      |z  z
  S r   )rJ   r   rq   r  r  r  s      r0   r   zgennorm_gen._logpdf*  s4    vvc$h"**SX"66QEEr2   c                     dt        j                  |      z  }d|z   |t        j                  d|z  t	        |      |z        z  z
  S r   )rJ   rK   rq   r  r  r?   rk   rg  r  s       r0   ro   zgennorm_gen._cdf*  s?    "''!*a1r||CHc!fdlCCCCr2   c                     t        j                  |dz
        }|t        j                  d|z  d|z   d|z  |z  z
        d|z  z  z  S )Nr   r   r   )rJ   rK   rq   r  r  s       r0   rx   zgennorm_gen._ppf*  sH    GGAG2??3t8cAgQq-@ACHMMMr2   c                 (    | j                  | |      S rH   r9  r  s      r0   rs   zgennorm_gen._sf*  s    yy!T""r2   c                 (    | j                  ||       S rH   rL  r  s      r0   r{   zgennorm_gen._isf*  s    		!T"""r2   c                     t        j                  d|z  d|z  d|z  g      \  }}}dt        j                  ||z
        dt        j                  ||z   d|z  z
        dz
  fS )Nr   r;  r  r   r   )rq   r  rJ   r   )r?   rg  c1c3c5s        r0   r   zgennorm_gen._stats*  s_    ZZT3t8SX >?
B266"r'?BrBwR/?(@2(EEEr2   c                 p    d|z  t        j                  d|z        z
  t        j                  d|z        z   S rM  r3  r?   rg  s     r0   r   zgennorm_gen._entropy*  s0    Dy266"t),,rzz"t)/DDDr2   Nc                     |j                  d|z  |      }|d|z  z  }t        j                  |      }|j                  |j                        dk  }||    ||<   |S )Nr   r  r   )r  rJ   r   randomrn  )r?   rg  r   r   r  r^  rK	  s          r0   r   zgennorm_gen._rvs*  sg     qvD1!D&MJJqM"""036T7($r2   r   )r}   r~   r   r   re   rl   r   ro   rx   rs   r{   r   r   r   r   r2   r0   r  r  m*  s@    )TH-FD
N
##FE	r2   r  gennormc                   @    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zy
)halfgennorm_gena  The upper half of a generalized normal continuous random variable.

    %(before_notes)s

    See Also
    --------
    gennorm : generalized normal distribution
    expon : exponential distribution
    halfnorm : half normal distribution

    Notes
    -----
    The probability density function for `halfgennorm` is:

    .. math::

        f(x, \beta) = \frac{\beta}{\Gamma(1/\beta)} \exp(-|x|^\beta)

    for :math:`x, \beta > 0`. :math:`\Gamma` is the gamma function
    (`scipy.special.gamma`).

    `halfgennorm` takes ``beta`` as a shape parameter for :math:`\beta`.
    For :math:`\beta = 1`, it is identical to an exponential distribution.
    For :math:`\beta = 2`, it is identical to a half normal distribution
    (with ``scale=1/sqrt(2)``).

    References
    ----------

    .. [1] "Generalized normal distribution, Version 1",
           https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1

    %(example)s

    c                 @    t        dddt        j                  fd      gS r  rb   rd   s    r0   re   zhalfgennorm_gen._shape_info*  r  r2   c                 L    t        j                  | j                  ||            S rH   r  r  s      r0   rl   zhalfgennorm_gen._pdf*  s     vvdll1d+,,r2   c                 j    t        j                  |      t        j                  d|z        z
  ||z  z
  S r  r3  r  s      r0   r   zhalfgennorm_gen._logpdf*  s+    vvd|bjjT22QW<<r2   c                 :    t        j                  d|z  ||z        S r  r  r  s      r0   ro   zhalfgennorm_gen._cdf*  s    {{3t8QW--r2   c                 @    t        j                  d|z  |      d|z  z  S r  r  r  s      r0   rx   zhalfgennorm_gen._ppf*  s     ~~c$h*SX66r2   c                 :    t        j                  d|z  ||z        S r  r  r  s      r0   rs   zhalfgennorm_gen._sf*  s    ||CHag..r2   c                 @    t        j                  d|z  |      d|z  z  S r  r  r  s      r0   r{   zhalfgennorm_gen._isf +  s     s4x+c$h77r2   c                 j    d|z  t        j                  |      z
  t        j                  d|z        z   S r  r3  r  s     r0   r   zhalfgennorm_gen._entropy+  s+    4x"&&,&CH)===r2   Nr  r   r2   r0   r  r  *  s1    "FH-=.7/8>r2   r  halfgennormc                   L     e Zd ZdZd Zd Z fdZd Zd Zd Z	d Z
d	 Z xZS )
crystalball_gena  
    Crystalball distribution

    %(before_notes)s

    Notes
    -----
    The probability density function for `crystalball` is:

    .. math::

        f(x, \beta, m) =  \begin{cases}
                            N \exp(-x^2 / 2),  &\text{for } x > -\beta\\
                            N A (B - x)^{-m}  &\text{for } x \le -\beta
                          \end{cases}

    where :math:`A = (m / |\beta|)^m  \exp(-\beta^2 / 2)`,
    :math:`B = m/|\beta| - |\beta|` and :math:`N` is a normalisation constant.

    `crystalball` takes :math:`\beta > 0` and :math:`m > 1` as shape
    parameters.  :math:`\beta` defines the point where the pdf changes
    from a power-law to a Gaussian distribution.  :math:`m` is the power
    of the power-law tail.

    %(after_notes)s

    .. versionadded:: 0.19.0

    References
    ----------
    .. [1] "Crystal Ball Function",
           https://en.wikipedia.org/wiki/Crystal_Ball_function

    %(example)s
    c                     |dkD  |dkD  z  S )z@
        Shape parameter bounds are m > 1 and beta > 0.
        r   r   r   )r?   rg  r  s      r0   r]   zcrystalball_gen._argcheck.+  s     A$(##r2   c                     t        dddt        j                  fd      }t        dddt        j                  fd      }||gS )Nrg  Fr   r  r  r   rb   )r?   ibetaims      r0   re   zcrystalball_gen._shape_info4+  s<    651bff+~FUQK@r{r2   c                 &    t         |   |d      S )N)r   r  rF  r  r  s     r0   r  zcrystalball_gen._fitstart9+  s    w H 55r2   c                     d||z  |dz
  z  t        j                  |dz   dz        z  t        t        |      z  z   z  }d }d }|t	        || kD  |||f||      z  S )a`  
        Return PDF of the crystalball function.

                                            --
                                           | exp(-x**2 / 2),  for x > -beta
        crystalball.pdf(x, beta, m) =  N * |
                                           | A * (B - x)**(-m), for x <= -beta
                                            --
        r   r   rO   r   c                 :    t        j                  | dz   dz        S r  r.  rk   rg  r  s      r0   rhsz!crystalball_gen._pdf.<locals>.rhsJ+  s    661a4%!)$$r2   c                 l    ||z  |z  t        j                  |dz   dz        z  ||z  |z
  | z
  | z  z  S r   r.  r  s      r0   lhsz!crystalball_gen._pdf.<locals>.lhsM+  sF    tVaK"&&$'C"88tVd]Q&1"-. /r2   r&  rJ   r   r   r   r   r?   rk   rg  r  r  r  r  s          r0   rl   zcrystalball_gen._pdf=+  ss     1T6QqS>BFFD!G8c>$::401 2	%	/ :a4%i!T1EEEr2   c                     d||z  |dz
  z  t        j                  |dz   dz        z  t        t        |      z  z   z  }d }d }t        j                  |      t        || kD  |||f||      z   S )zH
        Return the log of the PDF of the crystalball function.
        r   r   rO   r   c                     | dz   dz  S r  r   r  s      r0   r  z$crystalball_gen._logpdf.<locals>.rhsZ+  s    qD57Nr2   c                     |t        j                  ||z        z  |dz  dz  z
  |t        j                  ||z  |z
  | z
        z  z
  S r  r+  r  s      r0   r  z$crystalball_gen._logpdf.<locals>.lhs]+  sF    RVVAdF^#dAgai/!BFF1T6D=1;L4M2MMMr2   r&  )rJ   r   r   r   r   r   r  s          r0   r   zcrystalball_gen._logpdfS+  s|     1T6QqS>BFFD!G8c>$::401 2		N vvay:a4%i!T1MMMr2   c                     d||z  |dz
  z  t        j                  |dz   dz        z  t        t        |      z  z   z  }d }d }|t	        || kD  |||f||      z  S )z8
        Return CDF of the crystalball function
        r   r   rO   r   c                     ||z  t        j                  |dz   dz        z  |dz
  z  t        t        |       t        |       z
  z  z   S NrO   r   r   rJ   r   r   r   r  s      r0   r  z!crystalball_gen._cdf.<locals>.rhsi+  sN    tVrvvtQwhn551=9Q<)TE2B#BCD Er2   c                 ~    ||z  |z  t        j                  |dz   dz        z  ||z  |z
  | z
  | dz   z  z  |dz
  z  S r  r.  r  s      r0   r  z!crystalball_gen._cdf.<locals>.lhsm+  sV    tVaK"&&$'C"88tVd]Q&1"Q$/034Q38 9r2   r&  r  r  s          r0   ro   zcrystalball_gen._cdfb+  st     1T6QqS>BFFD!G8c>$::401 2	E	9 :a4%i!T1EEEr2   c                 
   d||z  |dz
  z  t        j                  |dz   dz        z  t        t        |      z  z   z  }|||z  z  t        j                  |dz   dz        z  |dz
  z  }d }d }t	        ||k  |||f||      S )Nr   r   rO   r   c                     t        j                  |dz   dz        }||z  |z  |dz
  z  }d|t        t        |      z  z   z  }||z  |z
  |dz
  ||z  | z  z  |z  | z  |z  dd|z
  z  z  z
  S r  r  r  rg  r  eb2r  r  s         r0   ppf_lessz&crystalball_gen._ppf.<locals>.ppf_lessx+  s    &&$'!$C43!A#&A1{Yt_445AdFTM!eaf^+C/1!3q!A#w?@ Ar2   c                     t        j                  |dz   dz        }||z  |z  |dz
  z  }d|t        t        |      z  z   z  }t	        t        |       dt        z  | |z  |z
  z  z         S r  )rJ   r   r   r   r   r  s         r0   ppf_greaterz)crystalball_gen._ppf.<locals>.ppf_greater+  sp    &&$'!$C43!A#&A1{Yt_445AYu-;1q0IIJJr2   r&  r  )r?   r  rg  r  r  pbetar  r  s           r0   rx   zcrystalball_gen._ppfs+  s    1T6QqS>BFFD!G8c>$::401 2QtVrvvtQwhqj11QU;	A	K !e)aq\X+NNr2   c           	         d||z  |dz
  z  t        j                  |dz   dz        z  t        t        |      z  z   z  }d }|t	        |dz   |k  |||ft        j
                  |t         j                  g      t         j                        z  S )zR
        Returns the n-th non-central moment of the crystalball function.
        r   r   rO   r   c                    ||z  |z  t        j                  |dz   dz        z  }||z  |z
  }d| dz
  dz  z  t        j                  | dz   dz        z  dd| z  t        j                  | dz   dz  |dz  dz        z  z   z  }t        j
                  |j                        }t        | dz         D ]C  }|t        j                  | |      || |z
  z  z  d|z  z  ||z
  dz
  z  ||z  | |z   dz   z  z  z  }E ||z  |z   S )z
            Returns n-th moment. Defined only if n+1 < m
            Function cannot broadcast due to the loop over n
            rO   r   r   r   r  )	rJ   r   rq   r  r  r  rn  r  binom)r\   rg  r  r!  r"  r  r  r
  s           r0   r
  z*crystalball_gen._munp.<locals>.n_th_moment+  s   
 4!bffdAgX^44A$A!Sy>BHHac1W$552'BKK1aq1$EEEGC((399%C1q5\ 0AQqS1R!G;q1uqyI4A26A:./ 00 s7S= r2   rH  )rJ   r   r   r   r   rB  rL  rc   )r?   r\   rg  r  r  r
  s         r0   r   zcrystalball_gen._munp+  s     1T6QqS>BFFD!G8c>$::401 2	! :a!eai!T1 ll;

|L ff& & 	&r2   )r}   r~   r   r   r]   re   r  rl   r   ro   rx   r   r  r  s   @r0   r  r  
+  s5    "F$
6F,NF"O(&r2   r  crystalballzA Crystalball Function)r   longnamec                 @    t        j                  d| dz  dz        dz  S )a  
    Utility function for the argus distribution used in the pdf, sf and
    moment calculation.
    Note that for all x > 0:
    gammainc(1.5, x**2/2) = 2 * (_norm_cdf(x) - x * _norm_pdf(x) - 0.5).
    This can be verified directly by noting that the cdf of Gamma(1.5) can
    be written as erf(sqrt(x)) - 2*sqrt(x)*exp(-x)/sqrt(Pi).
    We use gammainc instead of the usual definition because it is more precise
    for small chi.
    r  rO   r  )r  s    r0   
_argus_phir  +  s"     ;;sCF1H%))r2   c                   D    e Zd ZdZd Zd Zd Zd Zd ZddZ	dd	Z
d
 Zy)	argus_gena  
    Argus distribution

    %(before_notes)s

    Notes
    -----
    The probability density function for `argus` is:

    .. math::

        f(x, \chi) = \frac{\chi^3}{\sqrt{2\pi} \Psi(\chi)} x \sqrt{1-x^2}
                     \exp(-\chi^2 (1 - x^2)/2)

    for :math:`0 < x < 1` and :math:`\chi > 0`, where

    .. math::

        \Psi(\chi) = \Phi(\chi) - \chi \phi(\chi) - 1/2

    with :math:`\Phi` and :math:`\phi` being the CDF and PDF of a standard
    normal distribution, respectively.

    `argus` takes :math:`\chi` as shape a parameter. Details about sampling
    from the ARGUS distribution can be found in [2]_.

    %(after_notes)s

    References
    ----------
    .. [1] "ARGUS distribution",
           https://en.wikipedia.org/wiki/ARGUS_distribution
    .. [2] Christoph Baumgarten "Random variate generation by fast numerical
           inversion in the varying parameter case." Research in Statistics,
           vol. 1, 2023, doi:10.1080/27684520.2023.2279060.

    .. versionadded:: 0.19.0

    %(example)s
    c                 @    t        dddt        j                  fd      gS )Nr  Fr   r  rb   rd   s    r0   re   zargus_gen._shape_info+      5%!RVVnEFFr2   c                 h   t        j                  d      5  d||z  z
  }dt        j                  |      z  t        z
  t        j                  t	        |            z
  }|t        j                  |      z   dt        j
                  | |z        z  z   |dz  |z  dz  z
  cd d d        S # 1 sw Y   y xY w)Nr2  r3  r   r  r   rO   )rJ   r5  r   r   r  r  )r?   rk   r  r^  r!  s        r0   r   zargus_gen._logpdf+  s    [[) 	Gac	A"&&+.
31HHArvvay=3rxx1~#55Q
QF	G 	G 	Gs   BB((B1c                 L    t        j                  | j                  ||            S rH   r  r?   rk   r  s      r0   rl   zargus_gen._pdf+  s    vvdll1c*++r2   c                 ,    d| j                  ||      z
  S r  r  r  s      r0   ro   zargus_gen._cdf+  s    TXXa%%%r2   c                 h    t        |t        j                  d|dz  z
        z        t        |      z  S r*  )r  rJ   r   r  s      r0   rs   zargus_gen._sf+  s,    #AqD 112Z_DDr2   Nc                 Z  	
 t        j                  |      }|j                  dk(  r| j                  |||      }nt	        |j
                  |      \  }	t        t        j                  |            }t        j                  |      }t        j                  |gdgdgg      

j                  sqt        	
fdt        t        |       d      D              }| j                  
d   ||      }|j                  |      ||<   
j                          
j                  sq|dk(  r|d   }|S )	Nr   )r  r   r  r  r  c              3   \   K   | ]#  }|   sj                   |   n
t        d        % y wrH   r  r  s     r0   rl  z!argus_gen._rvs.<locals>.<genexpr>+  r  r  r   r   )rJ   r   r   r  r   rn  r  r  r  r  r  r  r  r  rE  r  )r?   r  r   r   r  r  r  r  r  r  r  s            @@r0   r   zargus_gen._rvs+  s   jjo88q=""340< # >C #399d3GCRWWS\*J((4.CC5"/&0\N4B kk ;%*CI:q%9; ;$$RUz2> % @99S>C kk 2:b'C
r2   c                    t        t        j                  |            }t        t        j                  |            }t        j
                  |      }d}||z  }|dk  r| dz  }	||k  r||z
  }
|j                  |
      }|j                  |
      }|dz  }t        j                  |      |	|z  k  }t        j                  |      }|dkD  r(t        j                  d||   z
        }|||||z    ||z  }||k  rn8|dk  rt        j                  | dz        }||k  r||z
  }
|j                  |
      }|j                  |
      }dt        j                  |d|z
  z  |z         z  |z  }|dz  |z   dk  }t        j                  |      }|dkD  r(t        j                  d||   z         }|||||z    ||z  }||k  rns||k  rP||z
  }
|j                  d|
      }||dz  k  }t        j                  |      }|dkD  r||   ||||z    ||z  }||k  rPt        j                  dd|z  |z  z
        }t        j                  ||      S )	Nr   r   rO   r  rA  r   g?r  )r  rJ   r  r  r  r  r  r   r  r   r   r  rE  )r?   r  r  r   r  r  rk   r  r  r  r
  r  r  r  r  r  r  echir  s                      r0   r  zargus_gen._rvs_scalar,  sq   h r}}Z01 HHQK	Sy#:	Aa-	M ((a(0 ((a(0H&&)q1u,VVF^
>''!ai-0C<?AiZ!79+I a- CZ664%!)$Da-	M ((a(0 ((a(0tq1u~122T9 Q$(a-VVF^
>''!ai-0C<?AiZ!79+I a- a-	M //!/<tax-VVF^
><=fIAiZ!79+I a- AEDL()Azz!V$$r2   c                    t        j                  |t              }t        |      }t        j                  t         j
                  dz        |z  t        j                  d|dz  dz        z  |z  }t        j                  |      }|dkD  }||   }dd|dz  z  z
  |t        |      z  ||   z  z   ||<   ||    }g d}t        j                  ||      || <   |||dz  z
  d d fS )	Nr  r'  r   rO   r  g?r  )	g_1g־r   gWBar   gp|RH?r   gE'卡?r   g?)rJ   r   rA  r  r   r   rq   r  r
  r   r  )r?   r  r  r  r=  rK	  r  coefs           r0   r   zargus_gen._statsl,  s     jjE*oGGBEE!Gs"RVVAsAvax%883>mmC SyIAqDL1y|#3c$i#??D	JKZZa(TE
#1*dD((r2   r   )r}   r~   r   r   re   r   rl   ro   rs   r   r  r   r   r2   r0   r  r  +  s5    'PGG,&E0c%J)r2   r  arguszAn Argus Function)r   r  r   r   c                   h     e Zd ZdZej
                  Zdd fd
Zd Zd Zd Z	d Z
d	 Z fd
Z xZS )rv_histograma3  
    Generates a distribution given by a histogram.
    This is useful to generate a template distribution from a binned
    datasample.

    As a subclass of the `rv_continuous` class, `rv_histogram` inherits from it
    a collection of generic methods (see `rv_continuous` for the full list),
    and implements them based on the properties of the provided binned
    datasample.

    Parameters
    ----------
    histogram : tuple of array_like
        Tuple containing two array_like objects.
        The first containing the content of n bins,
        the second containing the (n+1) bin boundaries.
        In particular, the return value of `numpy.histogram` is accepted.

    density : bool, optional
        If False, assumes the histogram is proportional to counts per bin;
        otherwise, assumes it is proportional to a density.
        For constant bin widths, these are equivalent, but the distinction
        is important when bin widths vary (see Notes).
        If None (default), sets ``density=True`` for backwards compatibility,
        but warns if the bin widths are variable. Set `density` explicitly
        to silence the warning.

        .. versionadded:: 1.10.0

    Notes
    -----
    When a histogram has unequal bin widths, there is a distinction between
    histograms that are proportional to counts per bin and histograms that are
    proportional to probability density over a bin. If `numpy.histogram` is
    called with its default ``density=False``, the resulting histogram is the
    number of counts per bin, so ``density=False`` should be passed to
    `rv_histogram`. If `numpy.histogram` is called with ``density=True``, the
    resulting histogram is in terms of probability density, so ``density=True``
    should be passed to `rv_histogram`. To avoid warnings, always pass
    ``density`` explicitly when the input histogram has unequal bin widths.

    There are no additional shape parameters except for the loc and scale.
    The pdf is defined as a stepwise function from the provided histogram.
    The cdf is a linear interpolation of the pdf.

    .. versionadded:: 0.19.0

    Examples
    --------

    Create a scipy.stats distribution from a numpy histogram

    >>> import scipy.stats
    >>> import numpy as np
    >>> data = scipy.stats.norm.rvs(size=100000, loc=0, scale=1.5,
    ...                             random_state=123)
    >>> hist = np.histogram(data, bins=100)
    >>> hist_dist = scipy.stats.rv_histogram(hist, density=False)

    Behaves like an ordinary scipy rv_continuous distribution

    >>> hist_dist.pdf(1.0)
    0.20538577847618705
    >>> hist_dist.cdf(2.0)
    0.90818568543056499

    PDF is zero above (below) the highest (lowest) bin of the histogram,
    defined by the max (min) of the original dataset

    >>> hist_dist.pdf(np.max(data))
    0.0
    >>> hist_dist.cdf(np.max(data))
    1.0
    >>> hist_dist.pdf(np.min(data))
    7.7591907244498314e-05
    >>> hist_dist.cdf(np.min(data))
    0.0

    PDF and CDF follow the histogram

    >>> import matplotlib.pyplot as plt
    >>> X = np.linspace(-5.0, 5.0, 100)
    >>> fig, ax = plt.subplots()
    >>> ax.set_title("PDF from Template")
    >>> ax.hist(data, density=True, bins=100)
    >>> ax.plot(X, hist_dist.pdf(X), label='PDF')
    >>> ax.plot(X, hist_dist.cdf(X), label='CDF')
    >>> ax.legend()
    >>> fig.show()

    N)densityc                t   || _         || _        t        |      dk7  rt        d      t	        j
                  |d         | _        t	        j
                  |d         | _        t        | j                        dz   t        | j                        k7  rt        d      | j                  dd | j                  dd z
  | _        t	        j                  | j                  | j                  d          }|#|r!d}t        j                  |t        d	       d
}n |s| j                  | j                  z  | _        | j                  t        t	        j                  | j                  | j                  z              z  | _        t	        j                  | j                  | j                  z        | _        t	        j"                  d| j                  dg      | _        t	        j"                  d| j                   g      | _        | j                  d   x|d<   | _        | j                  d   x|d<   | _        t)        | T  |i | y)a5  
        Create a new distribution using the given histogram

        Parameters
        ----------
        histogram : tuple of array_like
            Tuple containing two array_like objects.
            The first containing the content of n bins,
            the second containing the (n+1) bin boundaries.
            In particular, the return value of np.histogram is accepted.
        density : bool, optional
            If False, assumes the histogram is proportional to counts per bin;
            otherwise, assumes it is proportional to a density.
            For constant bin widths, these are equivalent.
            If None (default), sets ``density=True`` for backward
            compatibility, but warns if the bin widths are variable. Set
            `density` explicitly to silence the warning.
        rO   z)Expected length 2 for parameter histogramr   r   zbNumber of elements in histogram content and histogram boundaries do not match, expected n and n+1.Nr  zjBin widths are not constant. Assuming `density=True`.Specify `density` explicitly to silence this warning.r  Tr   r   r   )
_histogram_densityr  r   rJ   r   _hpdf_hbins_hbin_widthsallcloser  r  r  rA  r  cumsum_hcdfhstackr   r   r;   rI  )r?   	histogramr  rA   r(  	bins_varyr   r  s          r0   rI  zrv_histogram.__init__,  s   & $y>QHIIZZ	!-
jj1.tzz?Q#dkk"22 3 4 4 !KKOdkk#2.>>D$5$5t7H7H7KLL	?yOGMM'>a@Gd&7&77DJZZ%tzzD<M<M/M(N"OO
YYtzzD,=,==>
YYTZZ56
YYTZZ01
#{{1~-sdf#{{2.sdf$)&)r2   c                 `    | j                   t        j                  | j                  |d         S )z&
        PDF of the histogram
        r  )side)r  rJ   searchsortedr  r   s     r0   rl   zrv_histogram._pdf-  s$     zz"//$++qwGHHr2   c                 X    t        j                  || j                  | j                        S )z3
        CDF calculated from the histogram
        )rJ   interpr  r  r   s     r0   ro   zrv_histogram._cdf-  s     yyDKK44r2   c                 X    t        j                  || j                  | j                        S )zC
        Percentile function calculated from the histogram
        )rJ   r   r  r  r   s     r0   rx   zrv_histogram._ppf-  s     yyDJJ44r2   c                     | j                   dd |dz   z  | j                   dd |dz   z  z
  |dz   z  }t        j                  | j                  dd |z        S )z$Compute the n-th non-central moment.r   Nr  )r  rJ   r  r  )r?   r\   	integralss      r0   r   zrv_histogram._munp -  s[    [[_qs+dkk#2.>1.EE!A#N	vvdjj2&233r2   c                     t        | j                  dd dkD  | j                  dd ft        j                  d      }t        j                  | j                  dd |z  | j
                  z         S )zCompute entropy of distributionr   r  r   )r   r  rJ   r   r  r  )r?   r  s     r0   r   zrv_histogram._entropy%-  sh    Ab)C/**Qr*, tzz!B'#-0A0AABBBr2   c                 `    t         |          }| j                  |d<   | j                  |d<   |S )zF
        Set the histogram as additional constructor argument
        r  r  )r;   _updated_ctor_paramr  r  )r?   dctr  s     r0   r  z rv_histogram._updated_ctor_param--  s2     g)+??KI
r2   )r}   r~   r   r   r   r  rI  rl   ro   rx   r   r   r  r  r  s   @r0   r  r  ,  sE    Zv "//M15 .*`I554
C r2   r  c                   @     e Zd ZdZd Zd Z fdZd Zd Zd Z	 xZ
S )studentized_range_genu  A studentized range continuous random variable.

    %(before_notes)s

    See Also
    --------
    t: Student's t distribution

    Notes
    -----
    The probability density function for `studentized_range` is:

    .. math::

         f(x; k, \nu) = \frac{k(k-1)\nu^{\nu/2}}{\Gamma(\nu/2)
                        2^{\nu/2-1}} \int_{0}^{\infty} \int_{-\infty}^{\infty}
                        s^{\nu} e^{-\nu s^2/2} \phi(z) \phi(sx + z)
                        [\Phi(sx + z) - \Phi(z)]^{k-2} \,dz \,ds

    for :math:`x ≥ 0`, :math:`k > 1`, and :math:`\nu > 0`.

    `studentized_range` takes ``k`` for :math:`k` and ``df`` for :math:`\nu`
    as shape parameters.

    When :math:`\nu` exceeds 100,000, an asymptotic approximation (infinite
    degrees of freedom) is used to compute the cumulative distribution
    function [4]_ and probability distribution function.

    %(after_notes)s

    References
    ----------

    .. [1] "Studentized range distribution",
           https://en.wikipedia.org/wiki/Studentized_range_distribution
    .. [2] Batista, Ben Dêivide, et al. "Externally Studentized Normal Midrange
           Distribution." Ciência e Agrotecnologia, vol. 41, no. 4, 2017, pp.
           378-389., doi:10.1590/1413-70542017414047716.
    .. [3] Harter, H. Leon. "Tables of Range and Studentized Range." The Annals
           of Mathematical Statistics, vol. 31, no. 4, 1960, pp. 1122-1147.
           JSTOR, www.jstor.org/stable/2237810. Accessed 18 Feb. 2021.
    .. [4] Lund, R. E., and J. R. Lund. "Algorithm AS 190: Probabilities and
           Upper Quantiles for the Studentized Range." Journal of the Royal
           Statistical Society. Series C (Applied Statistics), vol. 32, no. 2,
           1983, pp. 204-210. JSTOR, www.jstor.org/stable/2347300. Accessed 18
           Feb. 2021.

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.stats import studentized_range
    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots(1, 1)

    Calculate the first four moments:

    >>> k, df = 3, 10
    >>> mean, var, skew, kurt = studentized_range.stats(k, df, moments='mvsk')

    Display the probability density function (``pdf``):

    >>> x = np.linspace(studentized_range.ppf(0.01, k, df),
    ...                 studentized_range.ppf(0.99, k, df), 100)
    >>> ax.plot(x, studentized_range.pdf(x, k, df),
    ...         'r-', lw=5, alpha=0.6, label='studentized_range pdf')

    Alternatively, the distribution object can be called (as a function)
    to fix the shape, location and scale parameters. This returns a "frozen"
    RV object holding the given parameters fixed.

    Freeze the distribution and display the frozen ``pdf``:

    >>> rv = studentized_range(k, df)
    >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

    Check accuracy of ``cdf`` and ``ppf``:

    >>> vals = studentized_range.ppf([0.001, 0.5, 0.999], k, df)
    >>> np.allclose([0.001, 0.5, 0.999], studentized_range.cdf(vals, k, df))
    True

    Rather than using (``studentized_range.rvs``) to generate random variates,
    which is very slow for this distribution, we can approximate the inverse
    CDF using an interpolator, and then perform inverse transform sampling
    with this approximate inverse CDF.

    This distribution has an infinite but thin right tail, so we focus our
    attention on the leftmost 99.9 percent.

    >>> a, b = studentized_range.ppf([0, .999], k, df)
    >>> a, b
    0, 7.41058083802274

    >>> from scipy.interpolate import interp1d
    >>> rng = np.random.default_rng()
    >>> xs = np.linspace(a, b, 50)
    >>> cdf = studentized_range.cdf(xs, k, df)
    # Create an interpolant of the inverse CDF
    >>> ppf = interp1d(cdf, xs, fill_value='extrapolate')
    # Perform inverse transform sampling using the interpolant
    >>> r = ppf(rng.uniform(size=1000))

    And compare the histogram:

    >>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
    >>> ax.legend(loc='best', frameon=False)
    >>> plt.show()

    c                     |dkD  |dkD  z  S r@  r   )r?   r
  r  s      r0   r]   zstudentized_range_gen._argcheck-  s    A"q&!!r2   c                     t        dddt        j                  fd      }t        dddt        j                  fd      }||gS )Nr
  Fr   r  r  r   rb   )r?   r)  r  s      r0   re   z!studentized_range_gen._shape_info-  s<    UQK@uq"&&k>BCyr2   c                 &    t         |   |d      S )N)rO   r   rF  r  r  s     r0   r  zstudentized_range_gen._fitstart-  s    w F 33r2   c                     d| j                         \  fd}t        j                  |dd      }t        j                   ||||      t        j                        d   S )N_studentized_range_momentc                    t        j                  ||      }| |||g}t        j                  |t              j
                  j                  t
        j                        }t        j                  t         |      }t        j                   t        j                  fdt        j                  f	
fg}t        dd      }t        j                  |||      d   S )Nr   r  -q=rP  rO  rangesopts)r   _studentized_range_pdf_logconstrJ   rQ  rA  rR  rS  rT  r   rU  rc   dictr   nquad)rF  r
  r  	log_constargusr_datar\  r  r  r  r  cython_symbols            r0   _single_momentz3studentized_range_gen._munp.<locals>._single_moment-  s    >>q"EIaY'CxxU+22::6??KH"..v}hOCw'!RVVr2h?FuU3D??3vDA!DDr2   r  r   r  r   )r   rJ   
frompyfuncr   rL  )	r?   rF  r
  r  r  ufuncr  r  r  s	         @@@r0   r   zstudentized_range_gen._munp-  sV    3""$B
	E na3zz%1b/<R@@r2   c                     d }t        j                  |dd      }t        j                   ||||      t         j                        d   S )Nc                    |dk  rd}t        j                  ||      }| |||g}t        j                  |t              j
                  j                  t
        j                        }t        j                   t        j                  fdt        j                  fg}nid}| |g}t        j                  |t              j
                  j                  t
        j                        }t        j                   t        j                  fg}t        j                  t         ||      }t        dd      }	t        j                  |||	      d   S )	N順 _studentized_range_pdfr   !_studentized_range_pdf_asymptoticr  r  r  r  )r   r  rJ   rQ  rA  rR  rS  rT  rc   r   rU  r  r   r  
rw   r
  r  r  r  r  r  r  r\  r  s
             r0   _single_pdfz/studentized_range_gen._pdf.<locals>._single_pdf-  s     F{ 8"BB1bI	!R+88C/66>>vOFF7BFF+a[9 !D!f88C/66>>vOFF7BFF+,"..v}hOCuU3D??3vDA!DDr2   r  r   r  r   )rJ   r  r   rL  )r?   rk   r
  r  r%  r  s         r0   rl   zstudentized_range_gen._pdf-  s>    	E( k1a0zz%1b/<R@@r2   c           	          d }t        j                  |dd      }t        j                  t        j                   ||||      t         j                        d   dd      S )Nc                    |dk  rd}t        j                  ||      }| |||g}t        j                  |t              j
                  j                  t
        j                        }t        j                   t        j                  fdt        j                  fg}nid}| |g}t        j                  |t              j
                  j                  t
        j                        }t        j                   t        j                  fg}t        j                  t         ||      }t        dd      }	t        j                  |||	      d   S )	Nr!  _studentized_range_cdfr   !_studentized_range_cdf_asymptoticr  r  r  r  )r   _studentized_range_cdf_logconstrJ   rQ  rA  rR  rS  rT  rc   r   rU  r  r   r  r$  s
             r0   _single_cdfz/studentized_range_gen._cdf.<locals>._single_cdf-  s    
 F{ 8"BB1bI	!R+88C/66>>vOFF7BFF+a[9 !D!f88C/66>>vOFF7BFF+,"..v}hOCuU3D??3vDA!DDr2   r  r   r  r   r   )rJ   r  r*	  r   rL  )r?   rk   r
  r  r+  r  s         r0   ro   zstudentized_range_gen._cdf-  sM    	E, k1a0 wwrzz%1b/DRH!QOOr2   )r}   r~   r   r   r]   re   r  r   rl   ro   r  r  s   @r0   r	  r	  7-  s+    l\"
4A*A2Pr2   r	  studentized_range)r   r   r   c                   \     e Zd ZdZd Zd Zd Zd Zd Zd Z	 e
e       fd       Z xZS )	rel_breitwigner_gena  A relativistic Breit-Wigner random variable.

    %(before_notes)s

    See Also
    --------
    cauchy: Cauchy distribution, also known as the Breit-Wigner distribution.

    Notes
    -----

    The probability density function for `rel_breitwigner` is

    .. math::

        f(x, \rho) = \frac{k}{(x^2 - \rho^2)^2 + \rho^2}

    where

    .. math::
        k = \frac{2\sqrt{2}\rho^2\sqrt{\rho^2 + 1}}
            {\pi\sqrt{\rho^2 + \rho\sqrt{\rho^2 + 1}}}

    The relativistic Breit-Wigner distribution is used in high energy physics
    to model resonances [1]_. It gives the uncertainty in the invariant mass,
    :math:`M` [2]_, of a resonance with characteristic mass :math:`M_0` and
    decay-width :math:`\Gamma`, where :math:`M`, :math:`M_0` and :math:`\Gamma`
    are expressed in natural units. In SciPy's parametrization, the shape
    parameter :math:`\rho` is equal to :math:`M_0/\Gamma` and takes values in
    :math:`(0, \infty)`.

    Equivalently, the relativistic Breit-Wigner distribution is said to give
    the uncertainty in the center-of-mass energy :math:`E_{\text{cm}}`. In
    natural units, the speed of light :math:`c` is equal to 1 and the invariant
    mass :math:`M` is equal to the rest energy :math:`Mc^2`. In the
    center-of-mass frame, the rest energy is equal to the total energy [3]_.

    %(after_notes)s

    :math:`\rho = M/\Gamma` and :math:`\Gamma` is the scale parameter. For
    example, if one seeks to model the :math:`Z^0` boson with :math:`M_0
    \approx 91.1876 \text{ GeV}` and :math:`\Gamma \approx 2.4952\text{ GeV}`
    [4]_ one can set ``rho=91.1876/2.4952`` and ``scale=2.4952``.

    To ensure a physically meaningful result when using the `fit` method, one
    should set ``floc=0`` to fix the location parameter to 0.

    References
    ----------
    .. [1] Relativistic Breit-Wigner distribution, Wikipedia,
           https://en.wikipedia.org/wiki/Relativistic_Breit-Wigner_distribution
    .. [2] Invariant mass, Wikipedia,
           https://en.wikipedia.org/wiki/Invariant_mass
    .. [3] Center-of-momentum frame, Wikipedia,
           https://en.wikipedia.org/wiki/Center-of-momentum_frame
    .. [4] M. Tanabashi et al. (Particle Data Group) Phys. Rev. D 98, 030001 -
           Published 17 August 2018

    %(example)s

    c                     |dkD  S r  r   r?   rhos     r0   r]   zrel_breitwigner_gen._argcheck@.  s    Qwr2   c                 @    t        dddt        j                  fd      gS )Nr1  Fr   r  rb   rd   s    r0   re   zrel_breitwigner_gen._shape_infoC.  r  r2   c           
      6   t        j                  ddd|dz  z  z   z  dt        j                  dd|dz  z  z         z   z        dz  t         j                  z  }t        j                  d      5  |||z
  ||z   z  |z  dz  dz   z  cd d d        S # 1 sw Y   y xY w)NrO   r   r2  rj  )rJ   r   r   r5  )r?   rk   r1  r  s       r0   rl   zrel_breitwigner_gen._pdfF.  s    GGQsAvX!bgga!CF(l&;";<
 [[h' 	:!c'AG,S014q89	: 	: 	:s   .BBc           
         t        j                  ddt        j                  dd|dz  z  z         z   z        t         j                  z  }t        j                  dd|z  z         t        j                  |t        j                  | |dz   z        z        z  }|dz  t        j                  |      z  }t        j
                  |d d      S )NrO   r   r  r
  )rJ   r   r   r}  imagr*	  )r?   rk   r1  r  r  s        r0   ro   zrel_breitwigner_gen._cdfN.  s    GGAq2771qax<00122558GGBCK ii"''3$b/2234 	 Q(wwvtQ''r2   c                 F   |dk(  rt        j                  ddd|dz  z  z   z  dt        j                  dd|dz  z  z         z   z        t         j                  z  |z  }|t         j                  dz  t        j                  |      z   z  S |dk(  rt        j                  dd|dz  z  z   ddt        j                  dd|dz  z  z         z   z  z        |z  }d|dz  z
  t        j                  dd|z  z
        z  }d|z  t        j                  |      z  S t         j
                  S )Nr   rO   r
  r  )rJ   r   r   r}  r  rc   )r?   r\   r1  r  r  s        r0   r   zrel_breitwigner_gen._munpY.  s   6Q36\"a"''!aQh,*?&?@A a"))C.0116QsAvX!q2771qax<+@'@"ABA #(lbggb2c6k&::Fq52776?**66Mr2   c                 F    d d t         j                  t         j                  fS rH   r  r0  s     r0   r   zrel_breitwigner_gen._statsj.  s     T266266))r2   c                    t        | |||      \  }}}}t        |t              }|r!|j                         dk(  r|j                  }d}||rt        |   |g|i |S |8t        j                  ||z
  g d      \  }}	}
|
|z
  }|	|z  }|s|g}d|vr(||d<   n"t        j                  ||z
        }||z  }|s|g}t        |   |g|i |S )Nr   F)r  r   g      ?r*   )
r  r9   r%   r:   r>   r;   r=   rJ   quantiler@  )r?   r@   rA   r/   r  r   r   rB   r  r  r  scale_0rho_0M_0r  s                 r0   r=   zrel_breitwigner_gen.fitp.  s    !<$d!
av dL1  "a' '' <87;t3d3d33> KKt5FGMCcCiG'MEwd" 'W))D4K(C&LEww{4/$/$//r2   )r}   r~   r   r   r]   re   rl   ro   r   r   r
   r   r=   r  r  s   @r0   r.  r.  .  sA    <zG:	("* M* 0 + 0r2   r.  rel_breitwignerrH   (@  r  collections.abcr   	functoolsr   r   rR  numpyrJ   numpy.polynomialr   scipy._lib.doccerr   r	   r
   scipy._lib._ccallbackr   scipyr   r   scipy.specialspecialrq   scipy.special._ufuncsr  rh   scipy._lib._utilr   r   rN  r   _tukeylambda_statsr   r@  r   rA  _distn_infrastructurer   r   r   r   r   r   r   _ksstatsr   r   r   
_constantsr   r   r    r!   r"   r#   r$   _censored_datar%   scipy.stats._boostr  rl  scipy.optimizer&   scipy.stats._warnings_errorsr'   scipy.statsr1   rE   rT   rV   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r  r  r-  r/  rB  r   rD  rQ   rX  r\  r^  rg  r  r  r  r  r  rQ  rS  rb  rd  ru  rw  r  r  r  r  r  r  r  r  r  r  r'  r)  r?  rD  r[  r_  ra  rq  rs  r  r  r  r  r  r  r|  r  r  r  r  r  r,  r.  r:  r<  rU  rW  r  r  r  r  r  r  r  r  r  r  r  r'  r)  r4  r6  r{  r}  r  r  r  r  r  r  r  r  r  r  r   r  r  r"  r$  r-  r/  rD  rF  re  rk  re  r  r  r  r  r  r  r  r  r  r0  r2  r5  rB  rU  r  rj  rx  rz  r  r  r  r  r  r  r  r  r  r  r  r  r  r%  r'  r4  r6  rq  rs  r  r  r  r  r  r  r  r  r  r  r  r  r%	  r?	  rA	  rP	  r]	  rh	  rj	  r	  r	  r	  r	  r	  r	  r	  r	  r	  r	  r	  r	  r
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  r,  r5  rE  rG  rJ  r  rU  rY  ru  rf  rL  r  r  r  r  r  r  r  r  r  r  r  r  r	  rc   r,  r.  r=  listglobalsrl	  itemspairs_distn_names_distn_gen_names__all__r   r2   r0   <module>rY     s  
  $ ,   '7 7 3    # # 4 BJ J J 2 1? ? ? ( # # & 1 9&*.9! 9!x 	C3W-<) <)@ 	!sc85M 5p Ck2	 bggag$+(k} k\ V3' 3'l 	Cg&( (V 
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	X 	f!} f!R #6*iM iX Ck2	4#= 4#n #:6~"} ~"B #F#Q, Q,h 
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Cc;= ;| #J/^Fm ^FB -8h!= h!V (-?@ D4m D4N -8Q?m Q?h =1uM up Ck2	=;= =;@ #J/R"] R"j .
)dM M` 	Cg&40 40n 
c	)^= ^B #J//%- /%d &2CDe eP "7->= ->` #J/*|= |~ Z(I= IX Z(W] Wt c5
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S= Sl #J/M!= M!` #J/zm zz	 -8P2} P2f  ^4@2] @2F al3
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@ {+	56M 56p C3[9	ZM Zz {+	E- EP 9
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c	)q7= q7h Z(o= od Z(^] ^@ c5
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} P
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c	)6! 6!r 	Cg&AD= ADH Z(W4= W4t #:640} 40n  #N;31M 31l {+	:3 :3z 	DCg.v= vr #J/a?] a?P .
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YF= YFx Z(gEM gET C3[9	 	s1==7EMF FR 
cSx	0<4] <4~ c5
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9%Pg7M g7T {A6	zm zz	 -8/.m /.d =1
L 
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%=>m =>@ 6U&m U&p =;ST*G) G)T 	w)<sKt= tnDPM DPN */Ba,.FF4 O0- O0d &2CD 	WY^^##%&!7}!M 
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