pywafo/wafo/stats/_continuous_distns.py

#
# Author:  Travis Oliphant  2002-2011 with contributions from
#          SciPy Developers 2004-2011
#
from __future__ import division, print_function, absolute_import

import warnings

from scipy.special import comb
from scipy.misc.doccer import inherit_docstring_from
from scipy import special
from scipy import optimize
from scipy import integrate
from scipy.special import (gammaln as gamln, gamma as gam, boxcox, boxcox1p,
                           log1p, expm1)  # inv_boxcox, inv_boxcox1p)

from numpy import (where, arange, putmask, ravel, sum, shape,
                   log, sqrt, exp, arctanh, tan, sin, arcsin, arctan,
                   tanh, cos, cosh, sinh)

from numpy import polyval, place, extract, any, asarray, nan, inf, pi

import numpy as np
import numpy.random as mtrand
try:
    from scipy.stats import vonmises_cython
except:
    vonmises_cython = None
# try:
#    from scipy.stats._tukeylambda_stats import \
#                        tukeylambda_variance as _tlvar, \
#                        tukeylambda_kurtosis as _tlkurt
# except:
#    _tlvar = _tlkurt = None
# from . import vonmises_cython
from ._tukeylambda_stats import (tukeylambda_variance as _tlvar,
                                 tukeylambda_kurtosis as _tlkurt)

from ._distn_infrastructure import (
    rv_continuous, valarray, _skew, _kurtosis, _lazywhere,
    _ncx2_log_pdf, _ncx2_pdf, _ncx2_cdf, get_distribution_names)

from ._constants import _XMIN, _EULER, _ZETA3, _EPS
from .stats import mode
# from .estimation import FitDistribution

__all__ = [
    'ksone', 'kstwobign', 'norm', 'alpha', 'anglit', 'arcsine',
    'beta', 'betaprime', 'bradford', 'burr', 'fisk', 'cauchy',
    'chi', 'chi2', 'cosine', 'dgamma', 'dweibull', 'erlang',
    'expon', 'exponweib', 'exponpow', 'fatiguelife', 'foldcauchy',
    'f', 'foldnorm', 'frechet_r', 'weibull_min', 'frechet_l',
    'weibull_max', 'genlogistic', 'genpareto', 'genexpon', 'genextreme',
    'gamma', 'gengamma', 'genhalflogistic', 'gompertz', 'gumbel_r',
    'gumbel_l', 'halfcauchy', 'halflogistic', 'halfnorm', 'hypsecant',
    'gausshyper', 'invgamma', 'invgauss', 'invweibull',
    'johnsonsb', 'johnsonsu', 'laplace', 'levy', 'levy_l',
    'levy_stable', 'logistic', 'loggamma', 'loglaplace', 'lognorm',
    'gilbrat', 'maxwell', 'mielke', 'nakagami', 'ncx2', 'ncf', 't',
    'nct', 'pareto', 'lomax', 'pearson3', 'powerlaw', 'powerlognorm',
    'powernorm', 'rdist', 'rayleigh', 'reciprocal', 'rice',
    'truncrayleigh',
    'recipinvgauss', 'semicircular', 'triang', 'truncexpon',
    'truncnorm', 'tukeylambda', 'uniform', 'vonmises', 'vonmises_line',
    'wald', 'wrapcauchy']


# Kolmogorov-Smirnov one-sided and two-sided test statistics
class ksone_gen(rv_continuous):

    """General Kolmogorov-Smirnov one-sided test.

    %(default)s

    """

    def _cdf(self, x, n):
        return 1.0 - special.smirnov(n, x)

    def _ppf(self, q, n):
        return special.smirnovi(n, 1.0 - q)
ksone = ksone_gen(a=0.0, name='ksone')


class kstwobign_gen(rv_continuous):

    """Kolmogorov-Smirnov two-sided test for large N.

    %(default)s

    """

    def _cdf(self, x):
        return 1.0 - special.kolmogorov(x)

    def _sf(self, x):
        return special.kolmogorov(x)

    def _ppf(self, q):
        return special.kolmogi(1.0 - q)
kstwobign = kstwobign_gen(a=0.0, name='kstwobign')


# Normal distribution

# loc = mu, scale = std
# Keep these implementations out of the class definition so they can be reused
# by other distributions.
_norm_pdf_C = np.sqrt(2 * pi)
_norm_pdf_logC = np.log(_norm_pdf_C)


def _norm_pdf(x):
    return exp(-x ** 2 / 2.0) / _norm_pdf_C


def _norm_logpdf(x):
    return -x ** 2 / 2.0 - _norm_pdf_logC


def _norm_cdf(x):
    return special.ndtr(x)


def _norm_logcdf(x):
    return special.log_ndtr(x)


def _norm_ppf(q):
    return special.ndtri(q)


def _norm_sf(x):
    return special.ndtr(-x)


def _norm_logsf(x):
    return special.log_ndtr(-x)


def _norm_isf(q):
    return -special.ndtri(q)


class norm_gen(rv_continuous):

    """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::

        norm.pdf(x) = exp(-x**2/2)/sqrt(2*pi)

    %(example)s

    """

    def _rvs(self):
        return mtrand.standard_normal(self._size)

    def _pdf(self, x):
        return _norm_pdf(x)

    def _logpdf(self, x):
        return _norm_logpdf(x)

    def _cdf(self, x):
        return _norm_cdf(x)

    def _logcdf(self, x):
        return _norm_logcdf(x)

    def _sf(self, x):
        return _norm_sf(x)

    def _logsf(self, x):
        return _norm_logsf(x)

    def _ppf(self, q):
        return _norm_ppf(q)

    def _isf(self, q):
        return _norm_isf(q)

    def _stats(self):
        return 0.0, 1.0, 0.0, 0.0

    def _entropy(self):
        return 0.5 * (log(2 * pi) + 1)

    @inherit_docstring_from(rv_continuous)
    def fit(self, data, **kwds):
        """%(super)s
        This function (norm_gen.fit) uses explicit formulas for the maximum
        likelihood estimation of the parameters, so the `optimizer` argument
        is ignored.
        """
        floc = kwds.get('floc', None)
        fscale = kwds.get('fscale', None)

        if floc is not None and fscale is not None:
            # This check is for consistency with `rv_continuous.fit`.
            # Without this check, this function would just return the
            # parameters that were given.
            raise ValueError("All parameters fixed. There is nothing to "
                             "optimize.")

        data = np.asarray(data)

        if floc is None:
            loc = data.mean()
        else:
            loc = floc

        if fscale is None:
            scale = np.sqrt(((data - loc) ** 2).mean())
        else:
            scale = fscale

        return loc, scale

norm = norm_gen(name='norm')


class alpha_gen(rv_continuous):

    """An alpha continuous random variable.

    %(before_notes)s

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

        alpha.pdf(x, a) = 1/(x**2*Phi(a)*sqrt(2*pi)) * exp(-1/2 * (a-1/x)**2),

    where ``Phi(alpha)`` is the normal CDF, ``x > 0``, and ``a > 0``.

    %(example)s

    """

    def _pdf(self, x, a):
        return 1.0 / (x ** 2) / special.ndtr(a) * _norm_pdf(a - 1.0 / x)

    def _logpdf(self, x, a):
        return -2 * log(x) + _norm_logpdf(a - 1.0 / x) - log(special.ndtr(a))

    def _cdf(self, x, a):
        return special.ndtr(a - 1.0 / x) / special.ndtr(a)

    def _ppf(self, q, a):
        return 1.0 / asarray(a - special.ndtri(q * special.ndtr(a)))

    def _stats(self, a):
        return [inf] * 2 + [nan] * 2
alpha = alpha_gen(a=0.0, name='alpha')


class anglit_gen(rv_continuous):

    """An anglit continuous random variable.

    %(before_notes)s

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

        anglit.pdf(x) = sin(2*x + pi/2) = cos(2*x),

    for ``-pi/4 <= x <= pi/4``.

    %(example)s

    """

    def _pdf(self, x):
        return cos(2 * x)

    def _cdf(self, x):
        return sin(x + pi / 4) ** 2.0

    def _ppf(self, q):
        return (arcsin(sqrt(q)) - pi / 4)

    def _stats(self):
        return 0.0, pi * pi / 16 - 0.5, 0.0, -2 * \
            (pi ** 4 - 96) / (pi * pi - 8) ** 2

    def _entropy(self):
        return 1 - log(2)
anglit = anglit_gen(a=-pi / 4, b=pi / 4, name='anglit')


class arcsine_gen(rv_continuous):

    """An arcsine continuous random variable.

    %(before_notes)s

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

        arcsine.pdf(x) = 1/(pi*sqrt(x*(1-x)))

    for ``0 < x < 1``.

    %(example)s

    """

    def _pdf(self, x):
        return 1.0 / pi / sqrt(x * (1 - x))

    def _cdf(self, x):
        return 2.0 / pi * arcsin(sqrt(x))

    def _ppf(self, q):
        return sin(pi / 2.0 * q) ** 2.0

    def _stats(self):
        mu = 0.5
        mu2 = 1.0 / 8
        g1 = 0
        g2 = -3.0 / 2.0
        return mu, mu2, g1, g2

    def _entropy(self):
        return -0.24156447527049044468
arcsine = arcsine_gen(a=0.0, b=1.0, name='arcsine')


class FitDataError(ValueError):
    # This exception is raised by, for example, beta_gen.fit when both floc
    # and fscale  are fixed and there are values in the data not in the open
    # interval (floc, floc+fscale).

    def __init__(self, distr, lower, upper):
        self.args = (
            "Invalid values in `data`.  Maximum likelihood "
            "estimation with {distr!r} requires that {lower!r} < x "
            "< {upper!r} for each x in `data`.".format(
                distr=distr, lower=lower, upper=upper),
        )


class FitSolverError(RuntimeError):
    # This exception is raised by, for example, beta_gen.fit when
    # optimize.fsolve returns with ier != 1.

    def __init__(self, mesg):
        emsg = "Solver for the MLE equations failed to converge: "
        emsg += mesg.replace('\n', '')
        self.args = (emsg,)


def _beta_mle_a(a, b, n, s1):
    # The zeros of this function give the MLE for `a`, with
    # `b`, `n` and `s1` given.  `s1` is the sum of the logs of
    # the data. `n` is the number of data points.
    psiab = special.psi(a + b)
    func = s1 - n * (-psiab + special.psi(a))
    return func


def _beta_mle_ab(theta, n, s1, s2):
    # Zeros of this function are critical points of
    # the maximum likelihood function.  Solving this system
    # for theta (which contains a and b) gives the MLE for a and b
    # given `n`, `s1` and `s2`.  `s1` is the sum of the logs of the data,
    # and `s2` is the sum of the logs of 1 - data.  `n` is the number
    # of data points.
    a, b = theta
    psiab = special.psi(a + b)
    func = [s1 - n * (-psiab + special.psi(a)),
            s2 - n * (-psiab + special.psi(b))]
    return func


class beta_gen(rv_continuous):

    """A beta continuous random variable.

    %(before_notes)s

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

                            gamma(a+b) * x**(a-1) * (1-x)**(b-1)
        beta.pdf(x, a, b) = ------------------------------------
                                     gamma(a)*gamma(b)

    for ``0 < x < 1``, ``a > 0``, ``b > 0``, where ``gamma(z)`` is the gamma
    function (`scipy.special.gamma`).

    %(example)s

    """

    def _rvs(self, a, b):
        return mtrand.beta(a, b, self._size)

    def _pdf(self, x, a, b):
        return np.exp(self._logpdf(x, a, b))

    def _logpdf(self, x, a, b):
        lPx = special.xlog1py(b - 1.0, -x) + special.xlogy(a - 1.0, x)
        lPx -= special.betaln(a, b)
        return lPx

    def _cdf(self, x, a, b):
        return special.btdtr(a, b, x)

    def _ppf(self, q, a, b):
        return special.btdtri(a, b, q)

    def _stats(self, a, b):
        mn = a * 1.0 / (a + b)
        var = (a * b * 1.0) / (a + b + 1.0) / (a + b) ** 2.0
        g1 = 2.0 * (b - a) * sqrt((1.0 + a + b) / (a * b)) / (2 + a + b)
        g2 = 6.0 * (a ** 3 + a ** 2 * (1 - 2 * b) + b **
                    2 * (1 + b) - 2 * a * b * (2 + b))
        g2 /= a * b * (a + b + 2) * (a + b + 3)
        return mn, var, g1, g2

    def _fitstart(self, data):
        g1 = _skew(data)
        g2 = _kurtosis(data)

        def func(x):
            a, b = x
            sk = 2 * (b - a) * sqrt(a + b + 1) / (a + b + 2) / sqrt(a * b)
            ku = a ** 3 - a ** 2 * \
                (2 * b - 1) + b ** 2 * (b + 1) - 2 * a * b * (b + 2)
            ku /= a * b * (a + b + 2) * (a + b + 3)
            ku *= 6
            return [sk - g1, ku - g2]
        a, b = optimize.fsolve(func, (1.0, 1.0))
        return super(beta_gen, self)._fitstart(data, args=(a, b))

    @inherit_docstring_from(rv_continuous)
    def fit(self, data, *args, **kwds):
        """%(super)s
        In the special case where both `floc` and `fscale` are given, a
        `ValueError` is raised if any value `x` in `data` does not satisfy
        `floc < x < floc + fscale`.
        """
        # Override rv_continuous.fit, so we can more efficiently handle the
        # case where floc and fscale are given.

        f0 = kwds.get('f0', None)
        f1 = kwds.get('f1', None)
        floc = kwds.get('floc', None)
        fscale = kwds.get('fscale', None)

        if floc is None or fscale is None:
            # do general fit
            return super(beta_gen, self).fit(data, *args, **kwds)

        if f0 is not None and f1 is not None:
            # This check is for consistency with `rv_continuous.fit`.
            raise ValueError("All parameters fixed. There is nothing to "
                             "optimize.")

        # Special case: loc and scale are constrained, so we are fitting
        # just the shape parameters.  This can be done much more efficiently
        # than the method used in `rv_continuous.fit`.  (See the subsection
        # "Two unknown parameters" in the section "Maximum likelihood" of
        # the Wikipedia article on the Beta distribution for the formulas.)

        # Normalize the data to the interval [0, 1].
        data = (ravel(data) - floc) / fscale
        if np.any(data <= 0) or np.any(data >= 1):
            raise FitDataError("beta", lower=floc, upper=floc + fscale)
        xbar = data.mean()

        if f0 is not None or f1 is not None:
            # One of the shape parameters is fixed.

            if f0 is not None:
                # The shape parameter a is fixed, so swap the parameters
                # and flip the data.  We always solve for `a`.  The result
                # will be swapped back before returning.
                b = f0
                data = 1 - data
                xbar = 1 - xbar
            else:
                b = f1

            # Initial guess for a.  Use the formula for the mean of the beta
            # distribution, E[x] = a / (a + b), to generate a reasonable
            # starting point based on the mean of the data and the given
            # value of b.
            a = b * xbar / (1 - xbar)

            # Compute the MLE for `a` by solving _beta_mle_a.
            theta, _info, ier, mesg = optimize.fsolve(
                _beta_mle_a, a, args=(b, len(data), np.log(data).sum()),
                full_output=True)
            if ier != 1:
                raise FitSolverError(mesg=mesg)
            a = theta[0]

            if f0 is not None:
                # The shape parameter a was fixed, so swap back the
                # parameters.
                a, b = b, a

        else:
            # Neither of the shape parameters is fixed.

            # s1 and s2 are used in the extra arguments passed to _beta_mle_ab
            # by optimize.fsolve.
            s1 = np.log(data).sum()
            s2 = np.log(1 - data).sum()

            # Use the "method of moments" to estimate the initial
            # guess for a and b.
            fac = xbar * (1 - xbar) / data.var(ddof=0) - 1
            a = xbar * fac
            b = (1 - xbar) * fac

            # Compute the MLE for a and b by solving _beta_mle_ab.
            theta, _info, ier, mesg = optimize.fsolve(
                _beta_mle_ab, [a, b], args=(len(data), s1, s2),
                full_output=True)
            if ier != 1:
                raise FitSolverError(mesg=mesg)
            a, b = theta

        return a, b, floc, fscale

beta = beta_gen(a=0.0, b=1.0, name='beta')


class betaprime_gen(rv_continuous):

    """A beta prime continuous random variable.

    %(before_notes)s

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

        betaprime.pdf(x, a, b) = x**(a-1) * (1+x)**(-a-b) / beta(a, b)

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

    %(example)s

    """

    def _rvs(self, a, b):
        u1 = gamma.rvs(a, size=self._size)
        u2 = gamma.rvs(b, size=self._size)
        return (u1 / u2)

    def _pdf(self, x, a, b):
        return np.exp(self._logpdf(x, a, b))

    def _logpdf(self, x, a, b):
        return (special.xlogy(a - 1.0, x) - special.xlog1py(a + b, x) -
                special.betaln(a, b))

    def _cdf(self, x, a, b):
        return special.betainc(a, b, x / (1. + x))
        # remove for now: special.hyp2f1 is incorrect for large a
        # x = where(x == 1.0, 1.0-1e-6, x)
        # return pow(x, a)*special.hyp2f1(a+b, a, 1+a, -x)/a/special.beta(a, b)

    def _fitstart(self, data):
        g1 = np.mean(data)
        g2 = mode(data)[0]

        def func(x):
            a, b = x
            me = a / (b - 1) if 1 < b else 1e100
            mo = (a - 1) / (b + 1) if 1 <= a else 0
            return [me - g1, mo - g2]
        a, b = optimize.fsolve(func, (1.0, 1.5))
        return super(betaprime_gen, self)._fitstart(data, args=(a, b))

    def _munp(self, n, a, b):
        if (n == 1.0):
            return where(b > 1, a / (b - 1.0), inf)
        elif (n == 2.0):
            return where(b > 2, a * (a + 1.0) / ((b - 2.0) * (b - 1.0)), inf)
        elif (n == 3.0):
            return where(b > 3,
                         a * (a + 1.0) * (a + 2.0) /
                         ((b - 3.0) * (b - 2.0) * (b - 1.0)),
                         inf)
        elif (n == 4.0):
            return where(b > 4,
                         a * (a + 1.0) * (a + 2.0) * (a + 3.0) /
                         ((b - 4.0) * (b - 3.0) * (b - 2.0) * (b - 1.0)),
                         inf)
        else:
            raise NotImplementedError
betaprime = betaprime_gen(a=0.0, name='betaprime')


class bradford_gen(rv_continuous):

    """A Bradford continuous random variable.

    %(before_notes)s

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

        bradford.pdf(x, c) = c / (k * (1+c*x)),

    for ``0 < x < 1``, ``c > 0`` and ``k = log(1+c)``.

    %(example)s

    """

    def _pdf(self, x, c):
        return c / (c * x + 1.0) / log1p(c)

    def _cdf(self, x, c):
        return log1p(c * x) / log1p(c)

    def _ppf(self, q, c):
        return ((1.0 + c) ** q - 1) / c

    def _stats(self, c, moments='mv'):
        k = log1p(c)
        mu = (c - k) / (c * k)
        mu2 = ((c + 2.0) * k - 2.0 * c) / (2 * c * k * k)
        g1 = None
        g2 = None
        if 's' in moments:
            g1 = (sqrt(2) * (12 * c * c - 9 * c * k * (c + 2) +
                             2 * k * k * (c * (c + 3) + 3)))
            g1 /= sqrt(c * (c * (k - 2) + 2 * k)) * (3 * c * (k - 2) + 6 * k)
        if 'k' in moments:
            g2 = (c ** 3 * (k - 3) * (k * (3 * k - 16) + 24) +
                  12 * k * c * c * (k - 4) * (k - 3)
                  + 6 * c * k * k * (3 * k - 14) + 12 * k ** 3)
            g2 /= 3 * c * (c * (k - 2) + 2 * k) ** 2
        return mu, mu2, g1, g2

    def _entropy(self, c):
        k = log1p(c)
        return k / 2.0 - log(c / k)

    def _fitstart(self, data):
        loc = data.min() - 1e-4
        scale = (data - loc).max()
        m = np.mean((data - loc) / scale)
        fun = lambda c: (c - log1p(c)) / (c * log1p(c)) - m
        res = optimize.root(fun, 0.3)
        c = res.x
        return c, loc, scale
bradford = bradford_gen(a=0.0, b=1.0, name='bradford')


class burr_gen(rv_continuous):

    """A Burr continuous random variable.

    %(before_notes)s

    See Also
    --------
    fisk : a special case of `burr` with ``d = 1``

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

        burr.pdf(x, c, d) = c * d * x**(-c-1) * (1+x**(-c))**(-d-1)

    for ``x > 0``.

    %(example)s

    """

    def _pdf(self, x, c, d):
        return c * d * (x ** (-c - 1.0)) * \
            ((1 + x ** (-c * 1.0)) ** (-d - 1.0))

    def _cdf(self, x, c, d):
        return (1 + x ** (-c * 1.0)) ** (-d ** 1.0)

    def _ppf(self, q, c, d):
        return (q ** (-1.0 / d) - 1) ** (-1.0 / c)

    def _munp(self, n, c, d):
        nc = 1. * n / c
        return d * special.beta(1.0 - nc, d + nc)
burr = burr_gen(a=0.0, name='burr')


class fisk_gen(burr_gen):

    """A Fisk continuous random variable.

    The Fisk distribution is also known as the log-logistic distribution, and
    equals the Burr distribution with ``d == 1``.

    %(before_notes)s

    See Also
    --------
    burr

    %(example)s

    """

    def _pdf(self, x, c):
        return burr_gen._pdf(self, x, c, 1.0)

    def _cdf(self, x, c):
        return burr_gen._cdf(self, x, c, 1.0)

    def _ppf(self, x, c):
        return burr_gen._ppf(self, x, c, 1.0)

    def _munp(self, n, c):
        return burr_gen._munp(self, n, c, 1.0)

    def _entropy(self, c):
        return 2 - log(c)
fisk = fisk_gen(a=0.0, name='fisk')


# median = loc
class cauchy_gen(rv_continuous):

    """A Cauchy continuous random variable.

    %(before_notes)s

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

        cauchy.pdf(x) = 1 / (pi * (1 + x**2))

    %(example)s

    """

    def _pdf(self, x):
        return 1.0 / pi / (1.0 + x * x)

    def _cdf(self, x):
        return 0.5 + 1.0 / pi * arctan(x)

    def _ppf(self, q):
        return tan(pi * q - pi / 2.0)

    def _sf(self, x):
        return 0.5 - 1.0 / pi * arctan(x)

    def _isf(self, q):
        return tan(pi / 2.0 - pi * q)

    def _stats(self):
        return inf, inf, nan, nan

    def _entropy(self):
        return log(4 * pi)

    def _fitstart(self, data, args=None):
        return (0, 1)
cauchy = cauchy_gen(name='cauchy')


class chi_gen(rv_continuous):

    """A chi continuous random variable.

    %(before_notes)s

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

        chi.pdf(x, df) = x**(df-1) * exp(-x**2/2) / (2**(df/2-1) * gamma(df/2))

    for ``x > 0``.

    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`

    %(example)s

    """

    def _rvs(self, df):
        return sqrt(chi2.rvs(df, size=self._size))

    def _pdf(self, x, df):
        return x ** (df - 1.) * exp(-x * x * 0.5) / \
            (2.0) ** (df * 0.5 - 1) / gam(df * 0.5)

    def _cdf(self, x, df):
        return special.gammainc(df * 0.5, 0.5 * x * x)

    def _ppf(self, q, df):
        return sqrt(2 * special.gammaincinv(df * 0.5, q))

    def _stats(self, df):
        mu = sqrt(2) * special.gamma(df / 2.0 + 0.5) / special.gamma(df / 2.0)
        mu2 = df - mu * mu
        g1 = (2 * mu ** 3.0 + mu * (1 - 2 * df)) / asarray(np.power(mu2, 1.5))
        g2 = 2 * df * (1.0 - df) - 6 * mu ** 4 + 4 * mu ** 2 * (2 * df - 1)
        g2 /= asarray(mu2 ** 2.0)
        return mu, mu2, g1, g2

    def _fitstart(self, data):
        m = data.mean()
        v = data.var()
        # Supply a starting guess with method of moments:
        df = max(np.round(v + m ** 2), 1)
        return super(chi_gen, self)._fitstart(data, args=(df,))
chi = chi_gen(a=0.0, name='chi')


# Chi-squared (gamma-distributed with loc=0 and scale=2 and shape=df/2)
class chi2_gen(rv_continuous):

    """A chi-squared continuous random variable.

    %(before_notes)s

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

        chi2.pdf(x, df) = 1 / (2*gamma(df/2)) * (x/2)**(df/2-1) * exp(-x/2)

    %(example)s

    """

    def _rvs(self, df):
        return mtrand.chisquare(df, self._size)

    def _pdf(self, x, df):
        return exp(self._logpdf(x, df))

    def _logpdf(self, x, df):
        return special.xlogy(df / 2. - 1, x) - x / 2. - \
            gamln(df / 2.) - (log(2) * df) / 2.

    def _cdf(self, x, df):
        return special.chdtr(df, x)

    def _sf(self, x, df):
        return special.chdtrc(df, x)

    def _isf(self, p, df):
        return special.chdtri(df, p)

    def _ppf(self, p, df):
        return self._isf(1.0 - p, df)

    def _stats(self, df):
        mu = df
        mu2 = 2 * df
        g1 = 2 * sqrt(2.0 / df)
        g2 = 12.0 / df
        return mu, mu2, g1, g2

    def _fitstart(self, data):
        m = data.mean()
        v = data.var()
        # Supply a starting guess with method of moments:
        df = max(np.round((m + v / 2) / 2), 1)
        return super(chi2_gen, self)._fitstart(data, args=(df,))
chi2 = chi2_gen(a=0.0, name='chi2')


class cosine_gen(rv_continuous):

    """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::

        cosine.pdf(x) = 1/(2*pi) * (1+cos(x))

    for ``-pi <= x <= pi``.

    %(example)s

    """

    def _pdf(self, x):
        return 1.0 / 2 / pi * (1 + cos(x))

    def _cdf(self, x):
        return 1.0 / 2 / pi * (pi + x + sin(x))

    def _stats(self):
        return 0.0, pi * pi / 3.0 - 2.0, 0.0, -6.0 * \
            (pi ** 4 - 90) / (5.0 * (pi * pi - 6) ** 2)

    def _entropy(self):
        return log(4 * pi) - 1.0
cosine = cosine_gen(a=-pi, b=pi, name='cosine')


class dgamma_gen(rv_continuous):

    """A double gamma continuous random variable.

    %(before_notes)s

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

        dgamma.pdf(x, a) = 1 / (2*gamma(a)) * abs(x)**(a-1) * exp(-abs(x))

    for ``a > 0``.

    %(example)s

    """

    def _rvs(self, a):
        u = mtrand.random_sample(size=self._size)
        return (gamma.rvs(a, size=self._size) * where(u >= 0.5, 1, -1))

    def _pdf(self, x, a):
        ax = abs(x)
        return 1.0 / (2 * special.gamma(a)) * ax ** (a - 1.0) * exp(-ax)

    def _logpdf(self, x, a):
        ax = abs(x)
        return special.xlogy(a - 1.0, ax) - ax - log(2) - gamln(a)

    def _cdf(self, x, a):
        fac = 0.5 * special.gammainc(a, abs(x))
        return where(x > 0, 0.5 + fac, 0.5 - fac)

    def _sf(self, x, a):
        fac = 0.5 * special.gammainc(a, abs(x))
        return where(x > 0, 0.5 - fac, 0.5 + fac)

    def _ppf(self, q, a):
        fac = special.gammainccinv(a, 1 - abs(2 * q - 1))
        return where(q > 0.5, fac, -fac)

    def _stats(self, a):
        mu2 = a * (a + 1.0)
        return 0.0, mu2, 0.0, (a + 2.0) * (a + 3.0) / mu2 - 3.0
dgamma = dgamma_gen(name='dgamma')


class dweibull_gen(rv_continuous):

    """A double Weibull continuous random variable.

    %(before_notes)s

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

        dweibull.pdf(x, c) = c / 2 * abs(x)**(c-1) * exp(-abs(x)**c)

    %(example)s

    """

    def _rvs(self, c):
        u = mtrand.random_sample(size=self._size)
        return weibull_min.rvs(c, size=self._size) * (where(u >= 0.5, 1, -1))

    def _pdf(self, x, c):
        ax = abs(x)
        Px = c / 2.0 * ax ** (c - 1.0) * exp(-ax ** c)
        return Px

    def _logpdf(self, x, c):
        ax = abs(x)
        return log(c) - log(2.0) + special.xlogy(c - 1.0, ax) - ax ** c

    def _cdf(self, x, c):
        Cx1 = 0.5 * exp(-abs(x) ** c)
        return where(x > 0, 1 - Cx1, Cx1)

    def _ppf(self, q, c):
        fac = 2. * where(q <= 0.5, q, 1. - q)
        fac = np.power(-log(fac), 1.0 / c)
        return where(q > 0.5, fac, -fac)

    def _munp(self, n, c):
        return (1 - (n % 2)) * special.gamma(1.0 + 1.0 * n / c)

    # since we know that all odd moments are zeros, return them at once.
    # returning Nones from _stats makes the public stats call _munp
    # so overall we're saving one or two gamma function evaluations here.
    def _stats(self, c):
        return 0, None, 0, None
dweibull = dweibull_gen(name='dweibull')


# Exponential (gamma distributed with a=1.0, loc=loc and scale=scale)
class expon_gen(rv_continuous):

    """An exponential continuous random variable.

    %(before_notes)s

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

        expon.pdf(x) = lambda * exp(- lambda*x)

    for ``x >= 0``.

    The scale parameter is equal to ``scale = 1.0 / lambda``.

    `expon` does not have shape parameters.

    %(example)s

    """

    def _link(self, x, logSF, phat, ix):
        if ix == 1:
            return - (x - phat[0]) / logSF
        elif ix == 0:
            return x + phat[1] * logSF
        else:
            raise IndexError('Index to the fixed parameter is out of bounds')

    def _rvs(self):
        return mtrand.standard_exponential(self._size)

    def _pdf(self, x):
        return exp(-x)

    def _logpdf(self, x):
        return -x

    def _cdf(self, x):
        return -expm1(-x)

    def _ppf(self, q):
        return -log1p(-q)

    def _sf(self, x):
        return exp(-x)

    def _logsf(self, x):
        return -x

    def _isf(self, q):
        return -log(q)

    def _stats(self):
        return 1.0, 1.0, 2.0, 6.0

    def _entropy(self):
        return 1.0
expon = expon_gen(a=0.0, name='expon')


class exponweib_gen(rv_continuous):

    """An exponentiated Weibull continuous random variable.

    %(before_notes)s

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

        exponweib.pdf(x, a, c) =
            a * c * (1-exp(-x**c))**(a-1) * exp(-x**c)*x**(c-1)

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

    %(example)s

    """

    def _pdf(self, x, a, c):
        return exp(self._logpdf(x, a, c))

    def _logpdf(self, x, a, c):
        negxc = -x ** c
        exm1c = -special.expm1(negxc)
        logp = (log(a) + log(c) + special.xlogy(a - 1.0, exm1c) +
                negxc + special.xlogy(c - 1.0, x))
        return logp

    def _cdf(self, x, a, c):
        exm1c = -expm1(-x ** c)
        return (exm1c) ** a

    def _ppf(self, q, a, c):
        return (-log1p(-q ** (1.0 / a))) ** asarray(1.0 / c)
exponweib = exponweib_gen(a=0.0, name='exponweib')


class exponpow_gen(rv_continuous):

    """An exponential power continuous random variable.

    %(before_notes)s

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

        exponpow.pdf(x, b) = b * x**(b-1) * exp(1 + x**b - exp(x**b))

    for ``x >= 0``, ``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".

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

    %(example)s

    """

    def _pdf(self, x, b):
        return exp(self._logpdf(x, b))

    def _logpdf(self, x, b):
        xb = x ** b
        f = 1 + log(b) + special.xlogy(b - 1.0, x) + xb - exp(xb)
        return f

    def _cdf(self, x, b):
        return -expm1(-expm1(x ** b))

    def _sf(self, x, b):
        return exp(-expm1(x ** b))

    def _isf(self, x, b):
        return (log1p(-log(x))) ** (1. / b)

    def _ppf(self, q, b):
        return pow(log1p(-log1p(-q)), 1.0 / b)
exponpow = exponpow_gen(a=0.0, name='exponpow')


class fatiguelife_gen(rv_continuous):

    """A fatigue-life (Birnbaum-Saunders) continuous random variable.

    %(before_notes)s

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

        fatiguelife.pdf(x, c) =
            (x+1) / (2*c*sqrt(2*pi*x**3)) * exp(-(x-1)**2/(2*x*c**2))

    for ``x > 0``.

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

    %(example)s

    """

    def _rvs(self, c):
        z = mtrand.standard_normal(self._size)
        x = 0.5 * c * z
        x2 = x * x
        t = 1.0 + 2 * x2 + 2 * x * sqrt(1 + x2)
        return t

    def _pdf(self, x, c):
        return np.exp(self._logpdf(x, c))

    def _logpdf(self, x, c):
        return (log(x + 1) - (x - 1) ** 2 / (2.0 * x * c ** 2) - log(2 * c) -
                0.5 * (log(2 * pi) + 3 * log(x)))

    def _cdf(self, x, c):
        return special.ndtr(1.0 / c * (sqrt(x) - 1.0 / sqrt(x)))

    def _ppf(self, q, c):
        tmp = c * special.ndtri(q)
        return 0.25 * (tmp + sqrt(tmp ** 2 + 4)) ** 2

    def _stats(self, c):
        # NB: the formula for kurtosis in wikipedia seems to have an error:
        # it's 40, not 41. At least it disagrees with the one from Wolfram
        # Alpha.  And the latter one, below, passes the tests, while the wiki
        # one doesn't So far I didn't have the guts to actually check the
        # coefficients from the expressions for the raw moments.
        c2 = c * c
        mu = c2 / 2.0 + 1.0
        den = 5.0 * c2 + 4.0
        mu2 = c2 * den / 4.0
        g1 = 4 * c * (11 * c2 + 6.0) / np.power(den, 1.5)
        g2 = 6 * c2 * (93 * c2 + 40.0) / den ** 2.0
        return mu, mu2, g1, g2
fatiguelife = fatiguelife_gen(a=0.0, name='fatiguelife')


class foldcauchy_gen(rv_continuous):

    """A folded Cauchy continuous random variable.

    %(before_notes)s

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

        foldcauchy.pdf(x, c) = 1/(pi*(1+(x-c)**2)) + 1/(pi*(1+(x+c)**2))

    for ``x >= 0``.

    %(example)s

    """

    def _rvs(self, c):
        return abs(cauchy.rvs(loc=c, size=self._size))

    def _pdf(self, x, c):
        return 1.0 / pi * (1.0 / (1 + (x - c) ** 2) + 1.0 / (1 + (x + c) ** 2))

    def _cdf(self, x, c):
        return 1.0 / pi * (arctan(x - c) + arctan(x + c))

    def _stats(self, c):
        return inf, inf, nan, nan
foldcauchy = foldcauchy_gen(a=0.0, name='foldcauchy')


class f_gen(rv_continuous):

    """An F continuous random variable.

    %(before_notes)s

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

                             df2**(df2/2) * df1**(df1/2) * x**(df1/2-1)
        F.pdf(x, df1, df2) = --------------------------------------------
                             (df2+df1*x)**((df1+df2)/2) * B(df1/2, df2/2)

    for ``x > 0``.

    %(example)s

    """

    def _rvs(self, dfn, dfd):
        return mtrand.f(dfn, dfd, self._size)

    def _pdf(self, x, dfn, dfd):
        return exp(self._logpdf(x, dfn, dfd))

    def _logpdf(self, x, dfn, dfd):
        n = 1.0 * dfn
        m = 1.0 * dfd
        lPx = m / 2 * log(m) + n / 2 * log(n) + (n / 2 - 1) * log(x)
        lPx -= ((n + m) / 2) * log(m + n * x) + special.betaln(n / 2, m / 2)
        return lPx

    def _cdf(self, x, dfn, dfd):
        return special.fdtr(dfn, dfd, x)

    def _sf(self, x, dfn, dfd):
        return special.fdtrc(dfn, dfd, x)

    def _ppf(self, q, dfn, dfd):
        return special.fdtri(dfn, dfd, q)

    def _stats(self, dfn, dfd):
        v1, v2 = 1. * dfn, 1. * dfd
        v2_2, v2_4, v2_6, v2_8 = v2 - 2., v2 - 4., v2 - 6., v2 - 8.

        mu = _lazywhere(
            v2 > 2, (v2, v2_2),
            lambda v2, v2_2: v2 / v2_2,
            np.inf)

        mu2 = _lazywhere(
            v2 > 4, (v1, v2, v2_2, v2_4),
            lambda v1, v2, v2_2, v2_4:
            2 * v2 * v2 * (v1 + v2_2) / (v1 * v2_2 ** 2 * v2_4),
            np.inf)

        g1 = _lazywhere(
            v2 > 6, (v1, v2_2, v2_4, v2_6),
            lambda v1, v2_2, v2_4, v2_6:
            (2 * v1 + v2_2) / v2_6 * sqrt(v2_4 / (v1 * (v1 + v2_2))),
            np.nan)
        g1 *= np.sqrt(8.)

        g2 = _lazywhere(
            v2 > 8, (g1, v2_6, v2_8),
            lambda g1, v2_6, v2_8: (8 + g1 * g1 * v2_6) / v2_8,
            np.nan)
        g2 *= 3. / 2.

        return mu, mu2, g1, g2

    def _fitstart(self, data):
        m = data.mean()
        v = data.var()
        # Supply a starting guess with method of moments:
        dfd = max(np.round(2 * m / (m - 1)), 5)
        dfn = max(
            np.round(2 * dfd * dfd * (dfd - 2) /
                     (v * (dfd - 4) * (dfd - 2) ** 2 - 2 * dfd * dfd)), 1)
        return super(f_gen, self)._fitstart(data, args=(dfn, dfd,))
f = f_gen(a=0.0, name='f')


# Folded Normal
# abs(Z) where (Z is normal with mu=L and std=S so that c=abs(L)/S)
##
# note: regress docs have scale parameter correct, but first parameter
# he gives is a shape parameter A = c * scale

# Half-normal is folded normal with shape-parameter c=0.

class foldnorm_gen(rv_continuous):

    """A folded normal continuous random variable.

    %(before_notes)s

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

        foldnormal.pdf(x, c) = sqrt(2/pi) * cosh(c*x) * exp(-(x**2+c**2)/2)

    for ``c >= 0``.

    %(example)s

    """

    def _argcheck(self, c):
        return (c >= 0)

    def _rvs(self, c):
        return abs(mtrand.standard_normal(self._size) + c)

    def _pdf(self, x, c):
        return _norm_pdf(x + c) + _norm_pdf(x - c)

    def _cdf(self, x, c):
        return special.ndtr(x - c) + special.ndtr(x + c) - 1.0

    def _stats(self, c):
        # Regina C. Elandt, Technometrics 3, 551 (1961)
        # http://www.jstor.org/stable/1266561
        #
        c2 = c * c
        expfac = np.exp(-0.5 * c2) / np.sqrt(2. * pi)

        mu = 2. * expfac + c * special.erf(c / sqrt(2))
        mu2 = c2 + 1 - mu * mu

        g1 = 2. * (mu * mu * mu - c2 * mu - expfac)
        g1 /= np.power(mu2, 1.5)

        g2 = c2 * (c2 + 6.) + 3 + 8. * expfac * mu
        g2 += (2. * (c2 - 3.) - 3. * mu ** 2) * mu ** 2
        g2 = g2 / mu2 ** 2.0 - 3.

        return mu, mu2, g1, g2
foldnorm = foldnorm_gen(a=0.0, name='foldnorm')


# Extreme Value Type II or Frechet
# (defined in Regress+ documentation as Extreme LB) as
# a limiting value distribution.
##
class frechet_r_gen(rv_continuous):

    """A Frechet right (or Weibull minimum) continuous random variable.

    %(before_notes)s

    See Also
    --------
    weibull_min : The same distribution as `frechet_r`.
    frechet_l, weibull_max

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

        frechet_r.pdf(x, c) = c * x**(c-1) * exp(-x**c)

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

    %(example)s

    """

    def _link(self, x, logSF, phat, ix):
        if ix == 0:
            phati = log(-logSF) / log((x - phat[1]) / phat[2])
        elif ix == 1:
            phati = x - phat[2] * (-logSF) ** (1. / phat[0])
        elif ix == 2:
            phati = (x - phat[1]) / (-logSF) ** (1. / phat[0])
        else:
            raise IndexError('Index to the fixed parameter is out of bounds')
        return phati

    def _pdf(self, x, c):
        return c * pow(x, c - 1) * exp(-pow(x, c))

    def _logpdf(self, x, c):
        return log(c) + special.xlogy(c - 1, x) - pow(x, c)

    def _cdf(self, x, c):
        return -expm1(-pow(x, c))

    def _ppf(self, q, c):
        return pow(-log1p(-q), 1.0 / c)

    def _munp(self, n, c):
        return special.gamma(1.0 + n * 1.0 / c)

    def _entropy(self, c):
        return -_EULER / c - log(c) + _EULER + 1

    def _fitstart(self, data):
        loc = data.min() - 0.01  # *np.std(data)
        chat = 1. / (6 ** (1 / 2) / pi * np.std(log(data - loc)))
        scale = np.mean((data - loc) ** chat) ** (1. / chat)
        return chat, loc, scale

frechet_r = frechet_r_gen(a=0.0, name='frechet_r')
weibull_min = frechet_r_gen(a=0.0, name='weibull_min')


class frechet_l_gen(rv_continuous):

    """A Frechet left (or Weibull maximum) continuous random variable.

    %(before_notes)s

    See Also
    --------
    weibull_max : The same distribution as `frechet_l`.
    frechet_r, weibull_min

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

        frechet_l.pdf(x, c) = c * (-x)**(c-1) * exp(-(-x)**c)

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

    %(example)s

    """

    def _pdf(self, x, c):
        return c * pow(-x, c - 1) * exp(-pow(-x, c))

    def _cdf(self, x, c):
        return exp(-pow(-x, c))

    def _ppf(self, q, c):
        return -pow(-log(q), 1.0 / c)

    def _munp(self, n, c):
        val = special.gamma(1.0 + n * 1.0 / c)
        if (int(n) % 2):
            sgn = -1
        else:
            sgn = 1
        return sgn * val

    def _entropy(self, c):
        return -_EULER / c - log(c) + _EULER + 1

    def _fitstart(self, data):
        loc = data.max() + 0.1 * np.std(data)
        chat = 1. / (6 ** (1 / 2) / pi * np.std(log(loc - data)))
        scale = np.mean((loc - data) ** chat) ** (1. / chat)
        return chat, loc, scale
frechet_l = frechet_l_gen(b=0.0, name='frechet_l')
weibull_max = frechet_l_gen(b=0.0, name='weibull_max')


class genlogistic_gen(rv_continuous):

    """A generalized logistic continuous random variable.

    %(before_notes)s

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

        genlogistic.pdf(x, c) = c * exp(-x) / (1 + exp(-x))**(c+1)

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

    %(example)s

    """

    def _pdf(self, x, c):
        return exp(self._logpdf(x, c))

    def _logpdf(self, x, c):
        return log(c) - x - (c + 1.0) * log1p(exp(-x))

    def _cdf(self, x, c):
        Cx = (1 + exp(-x)) ** (-c)
        return Cx

    def _ppf(self, q, c):
        vals = -log(pow(q, -1.0 / c) - 1)
        return vals

    def _stats(self, c):
        zeta = special.zeta
        mu = _EULER + special.psi(c)
        mu2 = pi * pi / 6.0 + zeta(2, c)
        g1 = -2 * zeta(3, c) + 2 * _ZETA3
        g1 /= np.power(mu2, 1.5)
        g2 = pi ** 4 / 15.0 + 6 * zeta(4, c)
        g2 /= mu2 ** 2.0
        return mu, mu2, g1, g2
genlogistic = genlogistic_gen(name='genlogistic')


def log1pxdx(x):
    '''Computes Log(1+x)/x
    '''
    xd = where((x == 0) | (x == inf), 1.0, x)  # avoid 0/0 or inf/inf
    y = where(x == 0, 1.0, log1p(x) / xd)
    return where(x == inf, 0.0, y)


class genpareto_gen(rv_continuous):

    """A generalized Pareto continuous random variable.

    %(before_notes)s

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

        genpareto.pdf(x, c) = (1 + c * x)**(-1 - 1/c)

    defined for ``x >= 0`` if ``c >=0``, and for
    ``0 <= x <= -1/c`` if ``c < 0``.

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

        genpareto.pdf(x, c=0) = exp(-x)

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

        genpareto.cdf(x, c=-1) = x

    %(example)s

    """

    def _link(self, x, logSF, phat, ix):
        # Reference
        # Stuart Coles (2004)
        # "An introduction to statistical modelling of extreme values".
        # Springer series in statistics
        c, loc, scale = phat
        if ix == 2:
            # Reorganizing w.r.t.scale, Eq. 4.13 and 4.14, pp 81 in
            # Coles (2004) gives
            #   link = -(x-loc)*c/expm1(-c*logSF)
            if c != 0.0:
                phati = (x - loc) * c / expm1(-c * logSF)
            else:
                phati = -(x - loc) / logSF
        elif ix == 1:
            if c != 0:
                phati = x + scale * expm1(c * logSF) / c
            else:
                phati = x + scale * logSF
        elif ix == 0:
            raise NotImplementedError(
                'link(x,logSF,phat,i) where i=0 is not implemented!')
        else:
            raise IndexError('Index to the fixed parameter is out of bounds')
        return phati

    def _argcheck(self, c):
        c = asarray(c)
        self.b = _lazywhere(c < 0, (c,),
                            lambda c: -1. / c, np.inf)
        return where(abs(c) == inf, False, True)

    def _pdf(self, x, c):
        return exp(self._logpdf(x, c))

    def _logpdf(self, x, c):
        return _lazywhere((x == x) & (c != 0), (x, c),
                          lambda x, c: -special.xlog1py(c + 1., c * x) / c,
                          -x)

    def _cdf(self, x, c):
        return - expm1(self._logsf(x, c))

    def _sf(self, x, c):
        return exp(self._logsf(x, c))

    def _logsf(self, x, c):
        return _lazywhere((x == x) & (c != 0), (x, c),
                          lambda x, c: -log1p(c * x) / c,
                          -x)

    def _ppf(self, q, c):
        return -boxcox1p(-q, -c)

    def _isf(self, q, c):
        return -boxcox(q, -c)

    def _fitstart(self, data):
        d = asarray(data)
        loc = d.min() - 0.01 * d.std()
        # moments estimator
        d1 = d - loc
        m = d1.mean()
        s = d1.std()

        shape = ((m / s) ** 2 - 1) / 2
        scale = m * ((m / s) ** 2 + 1) / 2
        return shape, loc, scale

    def hessian_nnlf(self, theta, x, eps=None):
        try:
            loc = theta[-2]
            scale = theta[-1]
            args = tuple(theta[:-2])
        except IndexError:
            raise ValueError("Not enough input arguments.")
        if not self._argcheck(*args) or scale <= 0:
            return inf
        x = asarray((x - loc) / scale)
        cond0 = (x <= self.a) | (x >= self.b)
        if any(cond0):
            np = self.numargs + 2
            return valarray((np, np), value=nan)
        eps = _EPS
        c = args[0]
        n = len(x)
        if abs(c) > eps:
            cx = c * x
            sumlog1pcx = sum(log1p(cx))
            # LL = n*log(scale) + (1-1/k)*sumlog1mkxn
            r = x / (1.0 + cx)
            sumix = sum(1.0 / (1.0 + cx) ** 2.0)

            sumr = sum(r)
            sumr2 = sum(r ** 2.0)
            H11 = -2 * sumlog1pcx / c ** 3 + 2 * \
                sumr / c ** 2 + (1.0 + 1.0 / c) * sumr2
            H22 = c * (c + 1) * sumix / scale ** 2.0
            H33 = (n - 2 * (c + 1) * sumr +
                   c * (c + 1) * sumr2) / scale ** 2.0
            H12 = -sum((1 - x) / ((1 + cx) ** 2.0)) / scale
            H23 = -(c + 1) * sumix / scale ** 2.0
            H13 = -(sumr - (c + 1) * sumr2) / scale

        else:  # c == 0
            sumx = sum(x)
            # LL = n*log(scale) + sumx;

            sumx2 = sum(x ** 2.0)
            H11 = -(2 / 3) * sum(x ** 3.0) + sumx2
            H22 = 0.0
            H12 = -(n - sum(x)) / scale
            H23 = -n * 1.0 / scale ** 2.0
            H33 = (n - 2 * sumx) / scale ** 2.0
            H13 = -(sumx - sumx2) / scale

        # Hessian matrix
        H = [[H11, H12, H13], [H12, H22, H23], [H13, H23, H33]]
        return asarray(H)

    def __stats(self, c):
        # return None,None,None,None
        k = -c
        m = where(k < -1.0, inf, 1.0 / (1 + k))
        v = where(k < -0.5, nan, 1.0 / ((1 + k) ** 2.0 * (1 + 2 * k)))
        sk = where(k < -1.0 / 3, nan, 2. * (1 - k)
                   * sqrt(1 + 2.0 * k) / (1.0 + 3. * k))
        # E(X^r) = s^r*(-k)^-(r+1)*gamma(1+r)*gamma(-1/k-r)/gamma(1-1/k)
        #  = s^r*gamma(1+r)./( (1+k)*(1+2*k).*....*(1+r*k))
        # E[(1-k(X-m0)/s)^r] = 1/(1+k*r)

        # Ex3 = (sk.*sqrt(v)+3*m).*v+m^3
        # Ex3 = 6.*s.^3/((1+k).*(1+2*k).*(1+3*k))
        r = 4.0
        Ex4 = gam(1. + r) / \
            ((1. + k) * (1. + 2. * k) * (1. + 3. * k) * (1 + 4. * k))
        m1 = m
        ku = where(k < -1. / 4, nan,
                   (Ex4 - 4. * sk * v ** (3. / 2)
                    * m1 - 6 * m1 ** 2. * v - m1 ** 4.) / v ** 2. - 3.0)
        return m, v, sk, ku

    def _munp(self, n, c):
        def __munp(n, c):
            val = 0.0
            k = arange(0, n + 1)
            for ki, cnk in zip(k, comb(n, k)):
                val = val + cnk * (-1) ** ki / (1.0 - c * ki)
            return where(c * n < 1, val * (-1.0 / c) ** n, inf)
        return _lazywhere(c != 0, (c,),
                          lambda c: __munp(n, c),
                          gam(n + 1))

    def _entropy(self, c):
        return 1. + c
genpareto = genpareto_gen(a=0.0, name='genpareto')


class genexpon_gen(rv_continuous):

    """A generalized exponential continuous random variable.

    %(before_notes)s

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

        genexpon.pdf(x, a, b, c) = (a + b * (1 - exp(-c*x))) * \
                                   exp(-a*x - b*x + b/c * (1-exp(-c*x)))

    for ``x >= 0``, ``a, b, c > 0``.

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

     N. Balakrishnan, "The Exponential Distribution: Theory, Methods and
     Applications", Asit P. Basu.

    %(example)s

    """

    def _link(self, x, logSF, phat, ix):
        _a, b, c, loc, scale = phat
        xn = (x - loc) / scale
        fact1 = (xn + expm1(-c * xn) / c)
        if ix == 0:
            phati = b * fact1 + logSF
        elif ix == 1:
            phati = (phat[0] - logSF) / fact1
        elif ix in [2, 3, 4]:
            raise NotImplementedError('Only implemented for index in [0,1]!')
        else:
            raise IndexError('Index to the fixed parameter is out of bounds')

        return phati

    def _pdf(self, x, a, b, c):
        return (a + b * (-expm1(-c * x))) * exp((-a - b) * x +
                                                b * (-expm1(-c * x)) / c)

    def _cdf(self, x, a, b, c):
        return -expm1((-a - b) * x + b * (-expm1(-c * x)) / c)

    def _logpdf(self, x, a, b, c):
        return (np.log(a + b * (-expm1(-c * x))) + (-a - b) * x +
                b * (-expm1(-c * x)) / c)
genexpon = genexpon_gen(a=0.0, name='genexpon')


class genextreme_gen(rv_continuous):

    """A generalized extreme value continuous random variable.

    %(before_notes)s

    See Also
    --------
    gumbel_r

    Notes
    -----
    For ``c=0``, `genextreme` is equal to `gumbel_r`.
    The probability density function for `genextreme` is::

        genextreme.pdf(x, c) =
            exp(-exp(-x))*exp(-x),                    for c==0
            exp(-(1-c*x)**(1/c))*(1-c*x)**(1/c-1),    for x <= 1/c, c > 0

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

    %(example)s

    """

    def _argcheck(self, c):
        min = np.minimum  # @ReservedAssignment
        max = np.maximum  # @ReservedAssignment
        self.b = where(c > 0, 1.0 / max(c, _XMIN), inf)
        self.a = where(c < 0, 1.0 / min(c, -_XMIN), -inf)
        return where(abs(c) == inf, 0, 1)

    def _pdf(self, x, c):
        return exp(self._logpdf(x, c))

    def _logpdf(self, x, c):
        x1 = where((c == 0) & (x == inf), 0.0, x)
        cx = c * x1
        cond1 = (c == 0) * (x == x)
        logex2 = where(cond1, 0.0, log1p(-cx))
        logpex2 = -x * log1pxdx(-cx)
        # logpex2 = where(cond1,-x, logex2/c)
        pex2 = exp(logpex2)
        # Handle special cases
        logpdf = where(
            (cx == 1) | (cx == -inf), -inf, -pex2 + logpex2 - logex2)
        putmask(logpdf, (c == 1) & (x == 1), 0.0)
        return logpdf

    def _cdf(self, x, c):
        return exp(self._logcdf(x, c))

    def _logcdf(self, x, c):
        x1 = where((c == 0) & (x == inf), 0.0, x)
        cx = c * x1
        loglogcdf = -x * log1pxdx(-cx)
        # loglogcdf = where((c==0)*(x==x),-x,log1p(-cx)/c)
        return -exp(loglogcdf)

    def _sf(self, x, c):
        return -expm1(self._logcdf(x, c))

    def _ppf(self, q, c):
        x = -log(-log(q))
        return _lazywhere((x == x) & (c != 0), (x, c),
                          lambda x, c: -expm1(-c * x) / c, x)

    def _stats(self, c):
        g = lambda n: gam(n * c + 1)
        g1 = g(1)
        g2 = g(2)
        g3 = g(3)
        g4 = g(4)
        g2mg12 = where(abs(c) < 1e-7, (c * pi) ** 2.0 / 6.0, g2 - g1 ** 2.0)
        gam2k = where(abs(c) < 1e-7, pi ** 2.0 / 6.0,
                      expm1(gamln(2.0 * c + 1.0) -
                            2 * gamln(c + 1.0)) / c ** 2.0)
        eps = 1e-14
        gamk = where(abs(c) < eps, -_EULER, expm1(gamln(c + 1)) / c)

        m = where(c < -1.0, nan, -gamk)
        v = where(c < -0.5, nan, g1 ** 2.0 * gam2k)

        # skewness
        sk1 = where(c < -1. / 3, nan,
                    np.sign(c) * (-g3 + (g2 + 2 * g2mg12) * g1) /
                    ((g2mg12) ** (3. / 2.)))
        sk = where(abs(c) <= eps ** 0.29, 12 * sqrt(6) * _ZETA3 / pi ** 3, sk1)

        # kurtosis
        ku1 = where(c < -1. / 4, nan,
                    (g4 + (-4 * g3 + 3 * (g2 + g2mg12) * g1) * g1) /
                    ((g2mg12) ** 2))
        ku = where(abs(c) <= (eps) ** 0.23, 12.0 / 5.0, ku1 - 3.0)
        return m, v, sk, ku

    def _munp(self, n, c):
        k = arange(0, n + 1)
        vals = 1.0 / c ** n * sum(
            comb(n, k) * (-1) ** k * special.gamma(c * k + 1),
            axis=0)
        return where(c * n > -1, vals, inf)

    def _fitstart(self, data):
        d = asarray(data)
        # Probability weighted moments
        log = np.log
        n = len(d)
        d.sort()
        koeff1 = np.r_[0:n] / (n - 1)
        koeff2 = koeff1 * (np.r_[0:n] - 1) / (n - 2)
        b2 = np.dot(koeff2, d) / n
        b1 = np.dot(koeff1, d) / n
        b0 = d.mean()
        z = (2 * b1 - b0) / (3 * b2 - b0) - log(2) / log(3)
        shape = 7.8590 * z + 2.9554 * z ** 2
        scale = (2 * b1 - b0) * shape / \
            (exp(gamln(1 + shape)) * (1 - 2 ** (-shape)))
        loc = b0 + scale * (expm1(gamln(1 + shape))) / shape
        return shape, loc, scale
genextreme = genextreme_gen(name='genextreme')


def _digammainv(y):
    # 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 `digamma(x) - y = 0`.  There is probably room for
    # improvement, but currently it works over a wide range of y:
    #    >>> y = 64*np.random.randn(1000000)
    #    >>> y.min(), y.max()
    #    (-311.43592651416662, 351.77388222276869)
    #    x = [_digammainv(t) for t in y]
    #    np.abs(digamma(x) - y).max()
    #    1.1368683772161603e-13
    #
    _em = 0.5772156649015328606065120
    func = lambda x: special.digamma(x) - y
    if y > -0.125:
        x0 = exp(y) + 0.5
        if y < 10:
            # Some experimentation shows that newton reliably converges
            # must faster than fsolve in this y range.  For larger y,
            # newton sometimes fails to converge.
            value = optimize.newton(func, x0, tol=1e-10)
            return value
    elif y > -3:
        x0 = exp(y / 2.332) + 0.08661
    else:
        x0 = 1.0 / (-y - _em)

    value, _info, ier, _msg = optimize.fsolve(func, x0, xtol=1e-11,
                                              full_output=True)
    if ier != 1:
        raise RuntimeError("_digammainv: fsolve failed, y = %r" % y)

    return value[0]


# Gamma (Use MATLAB and MATHEMATICA (b=theta=scale, a=alpha=shape) definition)

# gamma(a, loc, scale)  with a an integer is the Erlang distribution
# gamma(1, loc, scale)  is the Exponential distribution
# gamma(df/2, 0, 2) is the chi2 distribution with df degrees of freedom.

class gamma_gen(rv_continuous):

    """A gamma continuous random variable.

    %(before_notes)s

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

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

        gamma.pdf(x, a) = lambda**a * x**(a-1) * exp(-lambda*x) / gamma(a)

    for ``x >= 0``, ``a > 0``. Here ``gamma(a)`` refers to the gamma function.

    The scale parameter is equal to ``scale = 1.0 / lambda``.

    `gamma` has a shape parameter `a` which needs to be set explicitly. For
    instance:

        >>> from scipy.stats import gamma
        >>> rv = gamma(3., loc = 0., scale = 2.)

    produces a frozen form of `gamma` with shape ``a = 3.``, ``loc =0.``
    and ``lambda = 1./scale = 1./2.``.

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

    %(example)s

    """

    def _rvs(self, a):
        return mtrand.standard_gamma(a, self._size)

    def _pdf(self, x, a):
        return exp(self._logpdf(x, a))

    def _logpdf(self, x, a):
        return special.xlogy(a - 1.0, x) - x - gamln(a)

    def _cdf(self, x, a):
        return special.gammainc(a, x)

    def _sf(self, x, a):
        return special.gammaincc(a, x)

    def _ppf(self, q, a):
        return special.gammaincinv(a, q)

    def _stats(self, a):
        return a, a, 2.0 / sqrt(a), 6.0 / a

    def _entropy(self, a):
        return special.psi(a) * (1 - a) + a + gamln(a)

    def _fitstart(self, data):
        # The skewness of the gamma distribution is `4 / sqrt(a)`.
        # We invert that to estimate the shape `a` using the skewness
        # of the data.  The formula is regularized with 1e-8 in the
        # denominator to allow for degenerate data where the skewness
        # is close to 0.
        a = 4 / (1e-8 + _skew(data) ** 2)
        return super(gamma_gen, self)._fitstart(data, args=(a,))

    @inherit_docstring_from(rv_continuous)
    def fit(self, data, *args, **kwds):
        f0 = kwds.get('f0', None)
        floc = kwds.get('floc', None)
        fscale = kwds.get('fscale', None)

        if floc is None:
            # loc is not fixed.  Use the default fit method.
            return super(gamma_gen, self).fit(data, *args, **kwds)

        # Special case: loc is fixed.

        if f0 is not None and fscale is not None:
            # This check is for consistency with `rv_continuous.fit`.
            # Without this check, this function would just return the
            # parameters that were given.
            raise ValueError("All parameters fixed. There is nothing to "
                             "optimize.")

        # Fixed location is handled by shifting the data.
        data = np.asarray(data)
        if np.any(data <= floc):
            raise FitDataError("gamma", lower=floc, upper=np.inf)
        if floc != 0:
            # Don't do the subtraction in-place, because `data` might be a
            # view of the input array.
            data = data - floc
        xbar = data.mean()

        # Three cases to handle:
        # * shape and scale both free
        # * shape fixed, scale free
        # * shape free, scale fixed

        if fscale is None:
            # scale is free
            if f0 is not None:
                # shape is fixed
                a = f0
            else:
                # shape and scale are both free.
                # The MLE for the shape parameter `a` is the solution to:
                # log(a) - special.digamma(a) - log(xbar) + log(data.mean) = 0
                s = log(xbar) - log(data).mean()
                func = lambda a: log(a) - special.digamma(a) - s
                aest = (3 - s + np.sqrt((s - 3) ** 2 + 24 * s)) / (12 * s)
                xa = aest * (1 - 0.4)
                xb = aest * (1 + 0.4)
                a = optimize.brentq(func, xa, xb, disp=0)

            # The MLE for the scale parameter is just the data mean
            # divided by the shape parameter.
            scale = xbar / a
        else:
            # scale is fixed, shape is free
            # The MLE for the shape parameter `a` is the solution to:
            # special.digamma(a) - log(data).mean() + log(fscale) = 0
            c = log(data).mean() - log(fscale)
            a = _digammainv(c)
            scale = fscale

        return a, floc, scale

gamma = gamma_gen(a=0.0, name='gamma')


class erlang_gen(gamma_gen):

    """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.

    """

    def _argcheck(self, a):
        allint = np.all(np.floor(a) == a)
        allpos = np.all(a > 0)
        if not allint:
            # An Erlang distribution shouldn't really have a non-integer
            # shape parameter, so warn the user.
            warnings.warn(
                'The shape parameter of the erlang distribution '
                'has been given a non-integer value %r.' % (a,),
                RuntimeWarning)
        return allpos

    def _fitstart(self, data):
        # Override gamma_gen_fitstart so that an integer initial value is
        # used.  (Also regularize the division, to avoid issues when
        # _skew(data) is 0 or close to 0.)
        a = int(4.0 / (1e-8 + _skew(data) ** 2))
        return super(gamma_gen, self)._fitstart(data, args=(a,))

    # Trivial override of the fit method, so we can monkey-patch its
    # docstring.
    def fit(self, data, *args, **kwds):
        return super(erlang_gen, self).fit(data, *args, **kwds)

    if fit.__doc__ is not None:
        fit.__doc__ = (rv_continuous.fit.__doc__ +
                       """
            Notes
            -----
            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.
            """)
erlang = erlang_gen(a=0.0, name='erlang')


class gengamma_gen(rv_continuous):

    """A generalized gamma continuous random variable.

    %(before_notes)s

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

        gengamma.pdf(x, a, c) = abs(c) * x**(c*a-1) * exp(-x**c) / gamma(a)

    for ``x > 0``, ``a > 0``, and ``c != 0``.

    %(example)s

    """

    def _argcheck(self, a, c):
        return (a > 0) & (c != 0)

    def _pdf(self, x, a, c):
        return exp(self._logpdf(x, a, c))

    def _logpdf(self, x, a, c):
        return log(abs(c)) + special.xlogy(c * a - 1, x) - x ** c - gamln(a)

    def _cdf(self, x, a, c):
        val = special.gammainc(a, x ** c)
        cond = c + 0 * val
        return where(cond > 0, val, 1 - val)

    def _ppf(self, q, a, c):
        val1 = special.gammaincinv(a, q)
        val2 = special.gammaincinv(a, 1.0 - q)
        ic = 1.0 / c
        cond = c + 0 * val1
        return where(cond > 0, val1 ** ic, val2 ** ic)

    def _munp(self, n, a, c):
        return special.gamma(a + n * 1.0 / c) / special.gamma(a)

    def _entropy(self, a, c):
        val = special.psi(a)
        return a * (1 - val) + 1.0 / c * val + gamln(a) - log(abs(c))
gengamma = gengamma_gen(a=0.0, name='gengamma')


class genhalflogistic_gen(rv_continuous):

    """A generalized half-logistic continuous random variable.

    %(before_notes)s

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

        genhalflogistic.pdf(x, c) = 2 * (1-c*x)**(1/c-1) / (1+(1-c*x)**(1/c))**2

    for ``0 <= x <= 1/c``, and ``c > 0``.

    %(example)s

    """

    def _argcheck(self, c):
        self.b = 1.0 / c
        return (c > 0)

    def _pdf(self, x, c):
        limit = 1.0 / c
        tmp = asarray(1 - c * x)
        tmp0 = tmp ** (limit - 1)
        tmp2 = tmp0 * tmp
        return 2 * tmp0 / (1 + tmp2) ** 2

    def _cdf(self, x, c):
        limit = 1.0 / c
        tmp = asarray(1 - c * x)
        tmp2 = tmp ** (limit)
        return (1.0 - tmp2) / (1 + tmp2)

    def _ppf(self, q, c):
        return 1.0 / c * (1 - ((1.0 - q) / (1.0 + q)) ** c)

    def _entropy(self, c):
        return 2 - (2 * c + 1) * log(2)
genhalflogistic = genhalflogistic_gen(a=0.0, name='genhalflogistic')


class gompertz_gen(rv_continuous):

    """A Gompertz (or truncated Gumbel) continuous random variable.

    %(before_notes)s

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

        gompertz.pdf(x, c) = c * exp(x) * exp(-c*(exp(x)-1))

    for ``x >= 0``, ``c > 0``.

    %(example)s

    """

    def _pdf(self, x, c):
        return exp(self._logpdf(x, c))

    def _logpdf(self, x, c):
        return log(c) + x - c * expm1(x)

    def _cdf(self, x, c):
        return -expm1(-c * expm1(x))

    def _ppf(self, q, c):
        return log1p(-1.0 / c * log1p(-q))

    def _entropy(self, c):
        return 1.0 - log(c) - exp(c) * special.expn(1, c)
gompertz = gompertz_gen(a=0.0, name='gompertz')


class gumbel_r_gen(rv_continuous):

    """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::

        gumbel_r.pdf(x) = exp(-(x + exp(-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.

    %(example)s

    """

    def _pdf(self, x):
        return exp(self._logpdf(x))

    def _logpdf(self, x):
        return -x - exp(-x)

    def _cdf(self, x):
        return exp(-exp(-x))

    def _logcdf(self, x):
        return -exp(-x)

    def _ppf(self, q):
        return -log(-log(q))

    def _stats(self):
        return _EULER, pi * pi / 6.0, 12 * sqrt(6) / pi ** 3 * _ZETA3, 12.0 / 5

    def _entropy(self):
        # http://en.wikipedia.org/wiki/Gumbel_distribution
        return _EULER + 1.
gumbel_r = gumbel_r_gen(name='gumbel_r')


class gumbel_l_gen(rv_continuous):

    """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::

        gumbel_l.pdf(x) = exp(x - exp(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.

    %(example)s

    """

    def _pdf(self, x):
        return exp(self._logpdf(x))

    def _logpdf(self, x):
        return x - exp(x)

    def _cdf(self, x):
        return -expm1(-exp(x))

    def _ppf(self, q):
        return log(-log1p(-q))

    def _stats(self):
        return -_EULER, pi * pi / 6.0, \
               -12 * sqrt(6) / pi ** 3 * _ZETA3, 12.0 / 5

    def _entropy(self):
        return _EULER + 1.
gumbel_l = gumbel_l_gen(name='gumbel_l')


class halfcauchy_gen(rv_continuous):

    """A Half-Cauchy continuous random variable.

    %(before_notes)s

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

        halfcauchy.pdf(x) = 2 / (pi * (1 + x**2))

    for ``x >= 0``.

    %(example)s

    """

    def _pdf(self, x):
        return 2.0 / pi / (1.0 + x * x)

    def _logpdf(self, x):
        return np.log(2.0 / pi) - special.log1p(x * x)

    def _cdf(self, x):
        return 2.0 / pi * arctan(x)

    def _ppf(self, q):
        return tan(pi / 2 * q)

    def _stats(self):
        return inf, inf, nan, nan

    def _entropy(self):
        return log(2 * pi)
halfcauchy = halfcauchy_gen(a=0.0, name='halfcauchy')


class halflogistic_gen(rv_continuous):

    """A half-logistic continuous random variable.

    %(before_notes)s

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

        halflogistic.pdf(x) = 2 * exp(-x) / (1+exp(-x))**2 = 1/2 * sech(x/2)**2

    for ``x >= 0``.

    %(example)s

    """

    def _pdf(self, x):
        return exp(self._logpdf(x))

    def _logpdf(self, x):
        return log(2) - x - 2. * special.log1p(exp(-x))

    def _cdf(self, x):
        return tanh(x / 2.0)

    def _ppf(self, q):
        return 2 * arctanh(q)

    def _munp(self, n):
        if n == 1:
            return 2 * log(2)
        if n == 2:
            return pi * pi / 3.0
        if n == 3:
            return 9 * _ZETA3
        if n == 4:
            return 7 * pi ** 4 / 15.0
        return 2 * (1 - pow(2.0, 1 - n)) * \
            special.gamma(n + 1) * special.zeta(n, 1)

    def _entropy(self):
        return 2 - log(2)
halflogistic = halflogistic_gen(a=0.0, name='halflogistic')


class halfnorm_gen(rv_continuous):

    """A half-normal continuous random variable.

    %(before_notes)s

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

        halfnorm.pdf(x) = sqrt(2/pi) * exp(-x**2/2)

    for ``x > 0``.

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

    %(example)s

    """

    def _rvs(self):
        return abs(mtrand.standard_normal(size=self._size))

    def _pdf(self, x):
        return sqrt(2.0 / pi) * exp(-x * x / 2.0)

    def _logpdf(self, x):
        return 0.5 * np.log(2.0 / pi) - x * x / 2.0

    def _cdf(self, x):
        return special.ndtr(x) * 2 - 1.0

    def _ppf(self, q):
        return special.ndtri((1 + q) / 2.0)

    def _stats(self):
        return (sqrt(2.0 / pi),
                1 - 2.0 / pi, sqrt(2) * (4 - pi) / (pi - 2) ** 1.5,
                8 * (pi - 3) / (pi - 2) ** 2)

    def _entropy(self):
        return 0.5 * log(pi / 2.0) + 0.5
halfnorm = halfnorm_gen(a=0.0, name='halfnorm')


class hypsecant_gen(rv_continuous):

    """A hyperbolic secant continuous random variable.

    %(before_notes)s

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

        hypsecant.pdf(x) = 1/pi * sech(x)

    %(example)s

    """

    def _pdf(self, x):
        return 1.0 / (pi * cosh(x))

    def _cdf(self, x):
        return 2.0 / pi * arctan(exp(x))

    def _ppf(self, q):
        return log(tan(pi * q / 2.0))

    def _stats(self):
        return 0, pi * pi / 4, 0, 2

    def _entropy(self):
        return log(2 * pi)
hypsecant = hypsecant_gen(name='hypsecant')


class gausshyper_gen(rv_continuous):

    """A Gauss hypergeometric continuous random variable.

    %(before_notes)s

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

        gausshyper.pdf(x, a, b, c, z) =
            C * x**(a-1) * (1-x)**(b-1) * (1+z*x)**(-c)

    for ``0 <= x <= 1``, ``a > 0``, ``b > 0``, and
    ``C = 1 / (B(a, b) F[2, 1](c, a; a+b; -z))``

    %(example)s

    """

    def _argcheck(self, a, b, c, z):
        return (a > 0) & (b > 0) & (c == c) & (z == z)

    def _pdf(self, x, a, b, c, z):
        Cinv = gam(a) * gam(b) / gam(a + b) * special.hyp2f1(c, a, a + b, -z)
        return 1.0 / Cinv * \
            x ** (a - 1.0) * (1.0 - x) ** (b - 1.0) / (1.0 + z * x) ** c

    def _munp(self, n, a, b, c, z):
        fac = special.beta(n + a, b) / special.beta(a, b)
        num = special.hyp2f1(c, a + n, a + b + n, -z)
        den = special.hyp2f1(c, a, a + b, -z)
        return fac * num / den
gausshyper = gausshyper_gen(a=0.0, b=1.0, name='gausshyper')


class invgamma_gen(rv_continuous):

    """An inverted gamma continuous random variable.

    %(before_notes)s

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

        invgamma.pdf(x, a) = x**(-a-1) / gamma(a) * exp(-1/x)

    for x > 0, a > 0.

    `invgamma` is a special case of `gengamma` with ``c == -1``.

    %(example)s

    """

    def _pdf(self, x, a):
        return exp(self._logpdf(x, a))

    def _logpdf(self, x, a):
        return (-(a + 1) * log(x) - gamln(a) - 1.0 / x)

    def _cdf(self, x, a):
        return 1.0 - special.gammainc(a, 1.0 / x)

    def _ppf(self, q, a):
        return 1.0 / special.gammaincinv(a, 1. - q)

    def _stats(self, a, moments='mvsk'):
        m1 = _lazywhere(a > 1, (a,), lambda x: 1. / (x - 1.), np.inf)
        m2 = _lazywhere(a > 2, (a,), lambda x: 1. / (x - 1.) ** 2 / (x - 2.),
                        np.inf)

        g1, g2 = None, None
        if 's' in moments:
            g1 = _lazywhere(
                a > 3, (a,),
                lambda x: 4. * np.sqrt(x - 2.) / (x - 3.), np.nan)
        if 'k' in moments:
            g2 = _lazywhere(
                a > 4, (a,),
                lambda x: 6. * (5. * x - 11.) / (x - 3.) / (x - 4.), np.nan)
        return m1, m2, g1, g2

    def _entropy(self, a):
        return a - (a + 1.0) * special.psi(a) + gamln(a)
invgamma = invgamma_gen(a=0.0, name='invgamma')


# scale is gamma from DATAPLOT and B from Regress
class invgauss_gen(rv_continuous):

    """An inverse Gaussian continuous random variable.

    %(before_notes)s

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

        invgauss.pdf(x, mu) = 1 / sqrt(2*pi*x**3) * exp(-(x-mu)**2/(2*x*mu**2))

    for ``x > 0``.

    When `mu` is too small, evaluating the cumulative density function will be
    inaccurate due to ``cdf(mu -> 0) = inf * 0``.
    NaNs are returned for ``mu <= 0.0028``.

    %(example)s

    """

    def _rvs(self, mu):
        return mtrand.wald(mu, 1.0, size=self._size)

    def _pdf(self, x, mu):
        return 1.0 / sqrt(2 * pi * x ** 3.0) * \
            exp(-1.0 / (2 * x) * ((x - mu) / mu) ** 2)

    def _logpdf(self, x, mu):
        return -0.5 * log(2 * pi) - 1.5 * log(x) - \
            ((x - mu) / mu) ** 2 / (2 * x)

    def _cdf(self, x, mu):
        fac = sqrt(1.0 / x)
        # Numerical accuracy for small `mu` is bad.  See #869.
        C1 = _norm_cdf(fac * (x - mu) / mu)
        C1 += exp(1.0 / mu) * _norm_cdf(-fac * (x + mu) / mu) * exp(1.0 / mu)
        return C1

    def _stats(self, mu):
        return mu, mu ** 3.0, 3 * sqrt(mu), 15 * mu
invgauss = invgauss_gen(a=0.0, name='invgauss')


class invweibull_gen(rv_continuous):

    """An inverted Weibull continuous random variable.

    %(before_notes)s

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

        invweibull.pdf(x, c) = c * x**(-c-1) * exp(-x**(-c))

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

    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

    """

    def _pdf(self, x, c):
        xc1 = np.power(x, -c - 1.0)
        xc2 = np.power(x, -c)
        xc2 = exp(-xc2)
        return c * xc1 * xc2

    def _cdf(self, x, c):
        xc1 = np.power(x, -c)
        return exp(-xc1)

    def _ppf(self, q, c):
        return np.power(-log(q), -1.0 / c)

    def _munp(self, n, c):
        return special.gamma(1 - n / c)

    def _entropy(self, c):
        return 1 + _EULER + _EULER / c - log(c)
invweibull = invweibull_gen(a=0, name='invweibull')


class johnsonsb_gen(rv_continuous):

    """A Johnson SB continuous random variable.

    %(before_notes)s

    See Also
    --------
    johnsonsu

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

        johnsonsb.pdf(x, a, b) = b / (x*(1-x)) * phi(a + b * log(x/(1-x)))

    for ``0 < x < 1`` and ``a, b > 0``, and ``phi`` is the normal pdf.

    %(example)s

    """

    def _argcheck(self, a, b):
        return (b > 0) & (a == a)

    def _pdf(self, x, a, b):
        trm = _norm_pdf(a + b * log(x / (1.0 - x)))
        return b * 1.0 / (x * (1 - x)) * trm

    def _cdf(self, x, a, b):
        return _norm_cdf(a + b * log(x / (1.0 - x)))

    def _ppf(self, q, a, b):
        return 1.0 / (1 + exp(-1.0 / b * (_norm_ppf(q) - a)))
johnsonsb = johnsonsb_gen(a=0.0, b=1.0, name='johnsonsb')


class johnsonsu_gen(rv_continuous):

    """A Johnson SU continuous random variable.

    %(before_notes)s

    See Also
    --------
    johnsonsb

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

        johnsonsu.pdf(x, a, b) = b / sqrt(x**2 + 1) *
                                 phi(a + b * log(x + sqrt(x**2 + 1)))

    for all ``x, a, b > 0``, and `phi` is the normal pdf.

    %(example)s

    """

    def _argcheck(self, a, b):
        return (b > 0) & (a == a)

    def _pdf(self, x, a, b):
        x2 = x * x
        trm = _norm_pdf(a + b * log(x + sqrt(x2 + 1)))
        return b * 1.0 / sqrt(x2 + 1.0) * trm

    def _cdf(self, x, a, b):
        return _norm_cdf(a + b * log(x + sqrt(x * x + 1)))

    def _ppf(self, q, a, b):
        return sinh((_norm_ppf(q) - a) / b)
johnsonsu = johnsonsu_gen(name='johnsonsu')


class laplace_gen(rv_continuous):

    """A Laplace continuous random variable.

    %(before_notes)s

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

        laplace.pdf(x) = 1/2 * exp(-abs(x))

    %(example)s

    """

    def _rvs(self):
        return mtrand.laplace(0, 1, size=self._size)

    def _pdf(self, x):
        return 0.5 * exp(-abs(x))

    def _cdf(self, x):
        return where(x > 0, 1.0 - 0.5 * exp(-x), 0.5 * exp(x))

    def _ppf(self, q):
        return where(q > 0.5, -log(2) - log1p(-q), log(2 * q))

    def _stats(self):
        return 0, 2, 0, 3

    def _entropy(self):
        return log(2) + 1
laplace = laplace_gen(name='laplace')


class levy_gen(rv_continuous):

    """A Levy continuous random variable.

    %(before_notes)s

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

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

        levy.pdf(x) = 1 / (x * sqrt(2*pi*x)) * exp(-1/(2*x))

    for ``x > 0``.

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

    %(example)s

    """

    def _pdf(self, x):
        return 1 / sqrt(2 * pi * x) / x * exp(-1 / (2 * x))

    def _cdf(self, x):
        # Equivalent to 2*norm.sf(sqrt(1/x))
        return special.erfc(sqrt(0.5 / x))

    def _ppf(self, q):
        # Equivalent to 1.0/(norm.isf(q/2)**2) or 0.5/(erfcinv(q)**2)
        val = -special.ndtri(q / 2)
        return 1.0 / (val * val)

    def _stats(self):
        return inf, inf, nan, nan
levy = levy_gen(a=0.0, name="levy")


class levy_l_gen(rv_continuous):

    """A left-skewed Levy continuous random variable.

    %(before_notes)s

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

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

        levy_l.pdf(x) = 1 / (abs(x) * sqrt(2*pi*abs(x))) * exp(-1/(2*abs(x)))

    for ``x < 0``.

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

    %(example)s

    """

    def _pdf(self, x):
        ax = abs(x)
        return 1 / sqrt(2 * pi * ax) / ax * exp(-1 / (2 * ax))

    def _cdf(self, x):
        ax = abs(x)
        return 2 * _norm_cdf(1 / sqrt(ax)) - 1

    def _ppf(self, q):
        val = _norm_ppf((q + 1.0) / 2)
        return -1.0 / (val * val)

    def _stats(self):
        return inf, inf, nan, nan
levy_l = levy_l_gen(b=0.0, name="levy_l")


class levy_stable_gen(rv_continuous):

    """A Levy-stable continuous random variable.

    %(before_notes)s

    See Also
    --------
    levy, levy_l

    Notes
    -----
    Levy-stable distribution (only random variates available -- ignore other
    docs)

    %(example)s

    """

    def _rvs(self, alpha, beta):
        sz = self._size
        TH = uniform.rvs(loc=-pi / 2.0, scale=pi, size=sz)
        W = expon.rvs(size=sz)
        if alpha == 1:
            return 2 / pi * (pi / 2 + beta * TH) * tan(TH) - beta * \
                log((pi / 2 * W * cos(TH)) / (pi / 2 + beta * TH))

        ialpha = 1.0 / alpha
        aTH = alpha * TH
        if beta == 0:
            return W / (cos(TH) / tan(aTH) + sin(TH)) * \
                ((cos(aTH) + sin(aTH) * tan(TH)) / W) ** ialpha

        val0 = beta * tan(pi * alpha / 2)
        th0 = arctan(val0) / alpha
        val3 = W / (cos(TH) / tan(alpha * (th0 + TH)) + sin(TH))
        res3 = val3 * ((cos(aTH) + sin(aTH) * tan(TH) - val0 *
                        (sin(aTH) - cos(aTH) * tan(TH))) / W) ** ialpha
        return res3

    def _argcheck(self, alpha, beta):
        if beta == -1:
            self.b = 0.0
        elif beta == 1:
            self.a = 0.0
        return (alpha > 0) & (alpha <= 2) & (beta <= 1) & (beta >= -1)

    def _pdf(self, x, alpha, beta):
        raise NotImplementedError
levy_stable = levy_stable_gen(name='levy_stable')


class logistic_gen(rv_continuous):

    """A logistic (or Sech-squared) continuous random variable.

    %(before_notes)s

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

        logistic.pdf(x) = exp(-x) / (1+exp(-x))**2

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

    %(example)s

    """

    def _rvs(self):
        return mtrand.logistic(size=self._size)

    def _pdf(self, x):
        return exp(self._logpdf(x))

    def _logpdf(self, x):
        return -x - 2. * special.log1p(exp(-x))

    def _cdf(self, x):
        return special.expit(x)

    def _ppf(self, q):
        return -log1p(-q) + log(q)

    def _stats(self):
        return 0, pi * pi / 3.0, 0, 6.0 / 5.0

    def _entropy(self):
        # http://en.wikipedia.org/wiki/Logistic_distribution
        return 2.0
logistic = logistic_gen(name='logistic')


class loggamma_gen(rv_continuous):

    """A log gamma continuous random variable.

    %(before_notes)s

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

        loggamma.pdf(x, c) = exp(c*x-exp(x)) / gamma(c)

    for all ``x, c > 0``.

    %(example)s

    """

    def _rvs(self, c):
        return log(mtrand.gamma(c, size=self._size))

    def _pdf(self, x, c):
        return exp(c * x - exp(x) - gamln(c))

    def _cdf(self, x, c):
        return special.gammainc(c, exp(x))

    def _ppf(self, q, c):
        return log(special.gammaincinv(c, q))

    def _stats(self, c):
        # See, for example, "A Statistical Study of Log-Gamma Distribution", by
        # Ping Shing Chan (thesis, McMaster University, 1993).
        mean = special.digamma(c)
        var = special.polygamma(1, c)
        skewness = special.polygamma(2, c) / np.power(var, 1.5)
        excess_kurtosis = special.polygamma(3, c) / (var * var)
        return mean, var, skewness, excess_kurtosis

loggamma = loggamma_gen(name='loggamma')


class loglaplace_gen(rv_continuous):

    """A log-Laplace continuous random variable.

    %(before_notes)s

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

        loglaplace.pdf(x, c) = c / 2 * x**(c-1),   for 0 < x < 1
                             = c / 2 * x**(-c-1),  for x >= 1

    for ``c > 0``.

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

    %(example)s

    """

    def _pdf(self, x, c):
        cd2 = c / 2.0
        c = where(x < 1, c, -c)
        return cd2 * x ** (c - 1)

    def _cdf(self, x, c):
        return where(x < 1, 0.5 * x ** c, 1 - 0.5 * x ** (-c))

    def _ppf(self, q, c):
        return where(
            q < 0.5, (2.0 * q) ** (1.0 / c), (2 * (1.0 - q)) ** (-1.0 / c))

    def _munp(self, n, c):
        return c ** 2 / (c ** 2 - n ** 2)

    def _entropy(self, c):
        return log(2.0 / c) + 1.0
loglaplace = loglaplace_gen(a=0.0, name='loglaplace')


def _lognorm_logpdf(x, s):
    return -log(x) ** 2 / (2 * s ** 2) + \
        np.where(x == 0, 0, -log(s * x * sqrt(2 * pi)))


class lognorm_gen(rv_continuous):

    """A lognormal continuous random variable.

    %(before_notes)s

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

        lognorm.pdf(x, s) = 1 / (s*x*sqrt(2*pi)) * exp(-1/2*(log(x)/s)**2)

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

    If ``log(x)`` is normally distributed with mean ``mu`` and variance
    ``sigma**2``, then ``x`` is log-normally distributed with shape parameter
    sigma and scale parameter ``exp(mu)``.

    %(example)s

    """

    def _rvs(self, s):
        return exp(s * mtrand.standard_normal(self._size))

    def _pdf(self, x, s):
        return exp(self._logpdf(x, s))

    def _logpdf(self, x, s):
        return _lognorm_logpdf(x, s)

    def _cdf(self, x, s):
        return _norm_cdf(log(x) / s)

    def _ppf(self, q, s):
        return exp(s * _norm_ppf(q))

    def _stats(self, s):
        p = exp(s * s)
        mu = sqrt(p)
        mu2 = p * (p - 1)
        g1 = sqrt((p - 1)) * (2 + p)
        g2 = np.polyval([1, 2, 3, 0, -6.0], p)
        return mu, mu2, g1, g2

    def _entropy(self, s):
        return 0.5 * (1 + log(2 * pi) + 2 * log(s))

    def _fitstart(self, data):
        scale = data.std()
        loc = data.min() - 0.001
        logd = log(data - loc)
        m = logd.mean()
        s = sqrt((logd ** 2).mean() - m ** 2)
        return s, loc, scale
lognorm = lognorm_gen(a=0.0, name='lognorm')


class gilbrat_gen(rv_continuous):

    """A Gilbrat continuous random variable.

    %(before_notes)s

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

        gilbrat.pdf(x) = 1/(x*sqrt(2*pi)) * exp(-1/2*(log(x))**2)

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

    %(example)s

    """

    def _rvs(self):
        return exp(mtrand.standard_normal(self._size))

    def _pdf(self, x):
        return exp(self._logpdf(x))

    def _logpdf(self, x):
        return _lognorm_logpdf(x, 1.0)

    def _cdf(self, x):
        return _norm_cdf(log(x))

    def _ppf(self, q):
        return exp(_norm_ppf(q))

    def _stats(self):
        p = np.e
        mu = sqrt(p)
        mu2 = p * (p - 1)
        g1 = sqrt((p - 1)) * (2 + p)
        g2 = np.polyval([1, 2, 3, 0, -6.0], p)
        return mu, mu2, g1, g2

    def _entropy(self):
        return 0.5 * log(2 * pi) + 0.5

    def _fitstart(self, data):
        scale = data.std()
        loc = data.min() - 0.001
        return loc, scale
gilbrat = gilbrat_gen(a=0.0, name='gilbrat')


class maxwell_gen(rv_continuous):

    """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::

        maxwell.pdf(x) = sqrt(2/pi)x**2 * exp(-x**2/2)

    for ``x > 0``.

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

    %(example)s
    """

    def _rvs(self):
        return chi.rvs(3.0, size=self._size)

    def _pdf(self, x):
        return sqrt(2.0 / pi) * x * x * exp(-x * x / 2.0)

    def _cdf(self, x):
        return special.gammainc(1.5, x * x / 2.0)

    def _ppf(self, q):
        return sqrt(2 * special.gammaincinv(1.5, q))

    def _stats(self):
        val = 3 * pi - 8
        return (2 * sqrt(2.0 / pi), 3 -
                8 / pi, sqrt(2) * (32 - 10 * pi) / val ** 1.5,
                (-12 * pi * pi + 160 * pi - 384) / val ** 2.0)

    def _entropy(self):
        return _EULER + 0.5 * log(2 * pi) - 0.5
maxwell = maxwell_gen(a=0.0, name='maxwell')


class mielke_gen(rv_continuous):

    """A Mielke's Beta-Kappa continuous random variable.

    %(before_notes)s

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

        mielke.pdf(x, k, s) = k * x**(k-1) / (1+x**s)**(1+k/s)

    for ``x > 0``.

    %(example)s

    """

    def _pdf(self, x, k, s):
        return k * x ** (k - 1.0) / (1.0 + x ** s) ** (1.0 + k * 1.0 / s)

    def _cdf(self, x, k, s):
        return x ** k / (1.0 + x ** s) ** (k * 1.0 / s)

    def _ppf(self, q, k, s):
        qsk = pow(q, s * 1.0 / k)
        return pow(qsk / (1.0 - qsk), 1.0 / s)
mielke = mielke_gen(a=0.0, name='mielke')


class nakagami_gen(rv_continuous):

    """A Nakagami continuous random variable.

    %(before_notes)s

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

        nakagami.pdf(x, nu) = 2 * nu**nu / gamma(nu) *
                              x**(2*nu-1) * exp(-nu*x**2)

    for ``x > 0``, ``nu > 0``.

    %(example)s

    """

    def _pdf(self, x, nu):
        return 2 * nu ** nu / \
            gam(nu) * (x ** (2 * nu - 1.0)) * exp(-nu * x * x)

    def _cdf(self, x, nu):
        return special.gammainc(nu, nu * x * x)

    def _ppf(self, q, nu):
        return sqrt(1.0 / nu * special.gammaincinv(nu, q))

    def _stats(self, nu):
        mu = gam(nu + 0.5) / gam(nu) / sqrt(nu)
        mu2 = 1.0 - mu * mu
        g1 = mu * (1 - 4 * nu * mu2) / 2.0 / nu / np.power(mu2, 1.5)
        g2 = -6 * mu ** 4 * nu + (8 * nu - 2) * mu ** 2 - 2 * nu + 1
        g2 /= nu * mu2 ** 2.0
        return mu, mu2, g1, g2
nakagami = nakagami_gen(a=0.0, name="nakagami")


class ncx2_gen(rv_continuous):

    """A non-central chi-squared continuous random variable.

    %(before_notes)s

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

        ncx2.pdf(x, df, nc) = exp(-(nc+df)/2) * 1/2 * (x/nc)**((df-2)/4)
                              * I[(df-2)/2](sqrt(nc*x))

    for ``x > 0``.

    %(example)s

    """

    def _rvs(self, df, nc):
        return mtrand.noncentral_chisquare(df, nc, self._size)

    def _logpdf(self, x, df, nc):
        return _ncx2_log_pdf(x, df, nc)

    def _pdf(self, x, df, nc):
        return _ncx2_pdf(x, df, nc)

    def _cdf(self, x, df, nc):
        return _ncx2_cdf(x, df, nc)

    def _ppf(self, q, df, nc):
        return special.chndtrix(q, df, nc)

    def _stats(self, df, nc):
        val = df + 2.0 * nc
        return (df + nc, 2 * val, sqrt(8) * (val + nc) / val ** 1.5,
                12.0 * (val + 2 * nc) / val ** 2.0)

    def _fitstart(self, data):
        m = data.mean()
        v = data.var()
        # Supply a starting guess with method of moments:
        nc = (v / 2 - m) / 2
        df = m - nc
        return super(ncx2_gen, self)._fitstart(data, args=(df, nc))
ncx2 = ncx2_gen(a=0.0, name='ncx2')


class ncf_gen(rv_continuous):

    """A non-central F distribution continuous random variable.

    %(before_notes)s

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

        ncf.pdf(x, df1, df2, nc) = exp(nc/2 + nc*df1*x/(2*(df1*x+df2))) *
                    df1**(df1/2) * df2**(df2/2) * x**(df1/2-1) *
                    (df2+df1*x)**(-(df1+df2)/2) *
                    gamma(df1/2)*gamma(1+df2/2) *
                    L^{v1/2-1}^{v2/2}(-nc*v1*x/(2*(v1*x+v2))) /
                    (B(v1/2, v2/2) * gamma((v1+v2)/2))

    for ``df1, df2, nc > 0``.

    %(example)s

    """

    def _rvs(self, dfn, dfd, nc):
        return mtrand.noncentral_f(dfn, dfd, nc, self._size)

    def _pdf_skip(self, x, dfn, dfd, nc):
        n1, n2 = dfn, dfd
        term = -nc / 2 + nc * n1 * x / \
            (2 * (n2 + n1 * x)) + gamln(n1 / 2.) + gamln(1 + n2 / 2.)
        term -= gamln((n1 + n2) / 2.0)
        Px = exp(term)
        Px *= n1 ** (n1 / 2) * n2 ** (n2 / 2) * x ** (n1 / 2 - 1)
        Px *= (n2 + n1 * x) ** (-(n1 + n2) / 2)
        Px *= special.assoc_laguerre(-
                                     nc *
                                     n1 *
                                     x /
                                     (2.0 *
                                      (n2 +
                                       n1 *
                                       x)), n2 /
                                     2, n1 /
                                     2 -
                                     1)
        Px /= special.beta(n1 / 2, n2 / 2)
        # This function does not have a return.  Drop it for now, the generic
        # function seems to work OK.

    def _cdf(self, x, dfn, dfd, nc):
        return special.ncfdtr(dfn, dfd, nc, x)

    def _ppf(self, q, dfn, dfd, nc):
        return special.ncfdtri(dfn, dfd, nc, q)

    def _munp(self, n, dfn, dfd, nc):
        val = (dfn * 1.0 / dfd) ** n
        term = gamln(n + 0.5 * dfn) + gamln(0.5 * dfd - n) - gamln(dfd * 0.5)
        val *= exp(-nc / 2.0 + term)
        val *= special.hyp1f1(n + 0.5 * dfn, 0.5 * dfn, 0.5 * nc)
        return val

    def _stats(self, dfn, dfd, nc):
        mu = where(dfd <= 2, inf, dfd / (dfd - 2.0) * (1 + nc * 1.0 / dfn))
        mu2 = where(dfd <= 4, inf, 2 * (dfd * 1.0 / dfn) ** 2.0 *
                    ((dfn + nc / 2.0) ** 2.0 + (dfn + nc) * (dfd - 2.0)) /
                    ((dfd - 2.0) ** 2.0 * (dfd - 4.0)))
        return mu, mu2, None, None
ncf = ncf_gen(a=0.0, name='ncf')


class t_gen(rv_continuous):

    """A Student's T continuous random variable.

    %(before_notes)s

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

                                       gamma((df+1)/2)
        t.pdf(x, df) = ---------------------------------------------------
                       sqrt(pi*df) * gamma(df/2) * (1+x**2/df)**((df+1)/2)

    for ``df > 0``.

    %(example)s

    """

    def _rvs(self, df):
        return mtrand.standard_t(df, size=self._size)

    def _pdf(self, x, df):
        r = asarray(df * 1.0)
        Px = exp(gamln((r + 1) / 2) - gamln(r / 2))
        Px /= sqrt(r * pi) * (1 + (x ** 2) / r) ** ((r + 1) / 2)
        return Px

    def _logpdf(self, x, df):
        r = df * 1.0
        lPx = gamln((r + 1) / 2) - gamln(r / 2)
        lPx -= 0.5 * log(r * pi) + (r + 1) / 2 * log(1 + (x ** 2) / r)
        return lPx

    def _cdf(self, x, df):
        return special.stdtr(df, x)

    def _sf(self, x, df):
        return special.stdtr(df, -x)

    def _ppf(self, q, df):
        return special.stdtrit(df, q)

    def _isf(self, q, df):
        return -special.stdtrit(df, q)

    def _stats(self, df):
        mu2 = where(df > 2, df / (df - 2.0), inf)
        g1 = where(df > 3, 0.0, nan)
        g2 = where(df > 4, 6.0 / (df - 4.0), nan)
        return 0, mu2, g1, g2
t = t_gen(name='t')


class nct_gen(rv_continuous):

    """A non-central Student's T continuous random variable.

    %(before_notes)s

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

                                          df**(df/2) * gamma(df+1)
      nct.pdf(x, df, nc) = ----------------------------------------------------
                           2**df*exp(nc**2/2) * (df+x**2)**(df/2) * gamma(df/2)

    for ``df > 0``.

    %(example)s

    """

    def _argcheck(self, df, nc):
        return (df > 0) & (nc == nc)

    def _rvs(self, df, nc):
        return (norm.rvs(loc=nc, size=self._size) * sqrt(df) /
                sqrt(chi2.rvs(df, size=self._size)))

    def _pdf(self, x, df, nc):
        n = df * 1.0
        nc = nc * 1.0
        x2 = x * x
        ncx2 = nc * nc * x2
        fac1 = n + x2
        trm1 = n / 2. * log(n) + gamln(n + 1)
        trm1 -= n * log(2) + nc * nc / 2. + (n / 2.) * \
            log(fac1) + gamln(n / 2.)
        Px = exp(trm1)
        valF = ncx2 / (2 * fac1)
        trm1 = sqrt(2) * nc * x * special.hyp1f1(n / 2 + 1, 1.5, valF)
        trm1 /= asarray(fac1 * special.gamma((n + 1) / 2))
        trm2 = special.hyp1f1((n + 1) / 2, 0.5, valF)
        trm2 /= asarray(sqrt(fac1) * special.gamma(n / 2 + 1))
        Px *= trm1 + trm2
        return Px

    def _cdf(self, x, df, nc):
        return special.nctdtr(df, nc, x)

    def _ppf(self, q, df, nc):
        return special.nctdtrit(df, nc, q)

    def _fitstart(self, data):
        me = np.mean(data)
        # g2 = mode(data)[0]
        sa = np.std(data)

        def func(df):
            return ((df - 2) * (4 * df - 1) -
                    (4 * df - 1) * df / (sa ** 2 + me ** 2) +
                    me ** 2 / (sa ** 2 + me ** 2) * (df * (4 * df - 1) -
                                                     3 * df))

        df0 = np.maximum(2 * sa / (sa - 1), 1)
        df = optimize.fsolve(func, df0)
        mu = me * (1 - 3 / (4 * df - 1))
        return super(nct_gen, self)._fitstart(data, args=(df, mu))

    def _stats(self, df, nc, moments='mv'):
        #
        # See D. Hogben, R.S. Pinkham, and M.B. Wilk,
        # 'The moments of the non-central t-distribution'
        # Biometrika 48, p. 465 (2961).
        # e.g. http://www.jstor.org/stable/2332772 (gated)
        #
        g1 = g2 = None

        gfac = gam(df / 2. - 0.5) / gam(df / 2.)
        c11 = sqrt(df / 2.) * gfac
        c20 = df / (df - 2.)
        c22 = c20 - c11 * c11
        mu = np.where(df > 1, nc * c11, np.inf)
        mu2 = np.where(df > 2, c22 * nc * nc + c20, np.inf)
        if 's' in moments:
            c33t = df * (7. - 2. * df) / (df - 2.) / (df - 3.) + 2. * c11 * c11
            c31t = 3. * df / (df - 2.) / (df - 3.)
            mu3 = (c33t * nc * nc + c31t) * c11 * nc
            g1 = np.where(df > 3, mu3 / np.power(mu2, 1.5), np.nan)
        # kurtosis
        if 'k' in moments:
            c44 = df * df / (df - 2.) / (df - 4.)
            c44 -= c11 * c11 * 2. * df * (5. - df) / (df - 2.) / (df - 3.)
            c44 -= 3. * c11 ** 4
            c42 = df / (df - 4.) - c11 * c11 * (df - 1.) / (df - 3.)
            c42 *= 6. * df / (df - 2.)
            c40 = 3. * df * df / (df - 2.) / (df - 4.)

            mu4 = c44 * nc ** 4 + c42 * nc ** 2 + c40
            g2 = np.where(df > 4, mu4 / mu2 ** 2 - 3., np.nan)
        return mu, mu2, g1, g2
nct = nct_gen(name="nct")


class pareto_gen(rv_continuous):

    """A Pareto continuous random variable.

    %(before_notes)s

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

        pareto.pdf(x, b) = b / x**(b+1)

    for ``x >= 1``, ``b > 0``.

    %(example)s

    """

    def _pdf(self, x, b):
        return b * x ** (-b - 1)

    def _cdf(self, x, b):
        return 1 - x ** (-b)

    def _ppf(self, q, b):
        return pow(1 - q, -1.0 / b)

    def _stats(self, b, moments='mv'):
        mu, mu2, g1, g2 = None, None, None, None
        if 'm' in moments:
            mask = b > 1
            bt = extract(mask, b)
            mu = valarray(shape(b), value=inf)
            place(mu, mask, bt / (bt - 1.0))
        if 'v' in moments:
            mask = b > 2
            bt = extract(mask, b)
            mu2 = valarray(shape(b), value=inf)
            place(mu2, mask, bt / (bt - 2.0) / (bt - 1.0) ** 2)
        if 's' in moments:
            mask = b > 3
            bt = extract(mask, b)
            g1 = valarray(shape(b), value=nan)
            vals = 2 * (bt + 1.0) * sqrt(bt - 2.0) / ((bt - 3.0) * sqrt(bt))
            place(g1, mask, vals)
        if 'k' in moments:
            mask = b > 4
            bt = extract(mask, b)
            g2 = valarray(shape(b), value=nan)
            vals = (6.0 * polyval([1.0, 1.0, -6, -2], bt) /
                    polyval([1.0, -7.0, 12.0, 0.0], bt))
            place(g2, mask, vals)
        return mu, mu2, g1, g2

    def _entropy(self, c):
        return 1 + 1.0 / c - log(c)
pareto = pareto_gen(a=1.0, name="pareto")


class lomax_gen(rv_continuous):

    """A Lomax (Pareto of the second kind) continuous random variable.

    %(before_notes)s

    Notes
    -----
    The Lomax distribution is a special case of the Pareto distribution, with
    (loc=-1.0).

    The probability density function for `lomax` is::

        lomax.pdf(x, c) = c / (1+x)**(c+1)

    for ``x >= 0``, ``c > 0``.

    %(example)s

    """

    def _pdf(self, x, c):
        return c * 1.0 / (1.0 + x) ** (c + 1.0)

    def _logpdf(self, x, c):
        return log(c) - (c + 1) * log1p(x)

    def _cdf(self, x, c):
        return -expm1(-c * log1p(x))

    def _sf(self, x, c):
        return exp(-c * log1p(x))

    def _logsf(self, x, c):
        return -c * log1p(x)

    def _ppf(self, q, c):
        return expm1(-log1p(-q) / c)

    def _stats(self, c):
        mu, mu2, g1, g2 = pareto.stats(c, loc=-1.0, moments='mvsk')
        return mu, mu2, g1, g2

    def _entropy(self, c):
        return 1 + 1.0 / c - log(c)
lomax = lomax_gen(a=0.0, name="lomax")


class pearson3_gen(rv_continuous):

    """A pearson type III continuous random variable.

    %(before_notes)s

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

        pearson3.pdf(x, skew) = abs(beta) / gamma(alpha) *
            (beta * (x - zeta))**(alpha - 1) * exp(-beta*(x - zeta))

    where::

            beta = 2 / (skew * stddev)
            alpha = (stddev * beta)**2
            zeta = loc - alpha / beta

    %(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).

    """

    def _preprocess(self, x, skew):
        # The real 'loc' and 'scale' are handled in the calling pdf(...). The
        # local variables 'loc' and 'scale' within pearson3._pdf are set to
        # the defaults just to keep them as part of the equations for
        # documentation.
        loc = 0.0
        scale = 1.0

        # If skew is small, return _norm_pdf. The divide between pearson3
        # and norm was found by brute force and is approximately a skew of
        # 0.000016.  No one, I hope, would actually use a skew value even
        # close to this small.
        norm2pearson_transition = 0.000016

        ans, x, skew = np.broadcast_arrays([1.0], x, skew)
        ans = ans.copy()

        mask = np.absolute(skew) < norm2pearson_transition
        invmask = ~mask

        beta = 2.0 / (skew[invmask] * scale)
        alpha = (scale * beta) ** 2
        zeta = loc - alpha / beta

        transx = beta * (x[invmask] - zeta)
        return ans, x, transx, skew, mask, invmask, beta, alpha, zeta

    def _argcheck(self, skew):
        # The _argcheck function in rv_continuous only allows positive
        # arguments.  The skew argument for pearson3 can be zero (which I want
        # to handle inside pearson3._pdf) or negative.  So just return True
        # for all skew args.
        return np.ones(np.shape(skew), dtype=bool)

    def _stats(self, skew):
        # ans, x, transx, skew, mask, invmask, beta, alpha, zeta = (
        _1, _2, _3, skew, _4, _5, beta, alpha, zeta = self._preprocess([1],
                                                                       skew)
        m = zeta + alpha / beta
        v = alpha / (beta ** 2)
        s = 2.0 / (alpha ** 0.5) * np.sign(beta)
        k = 6.0 / alpha
        return m, v, s, k

    def _pdf(self, x, skew):
        # Do the calculation in _logpdf since helps to limit
        # overflow/underflow problems
        ans = exp(self._logpdf(x, skew))
        if ans.ndim == 0:
            if np.isnan(ans):
                return 0.0
            return ans
        ans[np.isnan(ans)] = 0.0
        return ans

    def _logpdf(self, x, skew):
        #   PEARSON3 logpdf                           GAMMA logpdf
        #   np.log(abs(beta))
        # + (alpha - 1)*log(beta*(x - zeta))          + (a - 1)*log(x)
        # - beta*(x - zeta)                           - x
        # - gamln(alpha)                              - gamln(a)
        ans, x, transx, skew, mask, invmask, beta, alpha, _zeta = (
            self._preprocess(x, skew))

        ans[mask] = np.log(_norm_pdf(x[mask]))
        ans[invmask] = log(abs(beta)) + gamma._logpdf(transx, alpha)
        return ans

    def _cdf(self, x, skew):
        ans, x, transx, skew, mask, invmask, _beta, alpha, _zeta = (
            self._preprocess(x, skew))

        ans[mask] = _norm_cdf(x[mask])
        ans[invmask] = gamma._cdf(transx, alpha)
        return ans

    def _rvs(self, skew):
        _ans, _x, _transx, skew, mask, _invmask, beta, alpha, zeta = (
            self._preprocess([0], skew))
        if mask[0]:
            return mtrand.standard_normal(self._size)
        ans = mtrand.standard_gamma(alpha, self._size) / beta + zeta
        if ans.size == 1:
            return ans[0]
        return ans

    def _ppf(self, q, skew):
        ans, q, _transq, skew, mask, invmask, beta, alpha, zeta = (
            self._preprocess(q, skew))
        ans[mask] = _norm_ppf(q[mask])
        ans[invmask] = special.gammaincinv(alpha, q[invmask]) / beta + zeta
        return ans
pearson3 = pearson3_gen(name="pearson3")


class powerlaw_gen(rv_continuous):

    """A power-function continuous random variable.

    %(before_notes)s

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

        powerlaw.pdf(x, a) = a * x**(a-1)

    for ``0 <= x <= 1``, ``a > 0``.

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

    %(example)s

    """

    def _pdf(self, x, a):
        return a * x ** (a - 1.0)

    def _logpdf(self, x, a):
        return log(a) + special.xlogy(a - 1, x)

    def _cdf(self, x, a):
        return x ** (a * 1.0)

    def _logcdf(self, x, a):
        return a * log(x)

    def _ppf(self, q, a):
        return pow(q, 1.0 / a)

    def _stats(self, a):
        return (a / (a + 1.0),
                a / (a + 2.0) / (a + 1.0) ** 2,
                -2.0 * ((a - 1.0) / (a + 3.0)) * sqrt((a + 2.0) / a),
                6 * polyval([1, -1, -6, 2], a) / (a * (a + 3.0) * (a + 4)))

    def _entropy(self, a):
        return 1 - 1.0 / a - log(a)
powerlaw = powerlaw_gen(a=0.0, b=1.0, name="powerlaw")


class powerlognorm_gen(rv_continuous):

    """A power log-normal continuous random variable.

    %(before_notes)s

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

        powerlognorm.pdf(x, c, s) = c / (x*s) * phi(log(x)/s) *
                                                (Phi(-log(x)/s))**(c-1),

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

    %(example)s

    """

    def _pdf(self, x, c, s):
        return (c / (x * s) * _norm_pdf(log(x) / s) *
                pow(_norm_cdf(-log(x) / s), c * 1.0 - 1.0))

    def _cdf(self, x, c, s):
        return 1.0 - pow(_norm_cdf(-log(x) / s), c * 1.0)

    def _ppf(self, q, c, s):
        return exp(-s * _norm_ppf(pow(1.0 - q, 1.0 / c)))
powerlognorm = powerlognorm_gen(a=0.0, name="powerlognorm")


class powernorm_gen(rv_continuous):

    """A power normal continuous random variable.

    %(before_notes)s

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

        powernorm.pdf(x, c) = c * phi(x) * (Phi(-x))**(c-1)

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

    %(example)s

    """

    def _pdf(self, x, c):
        return (c * _norm_pdf(x) * (_norm_cdf(-x) ** (c - 1.0)))

    def _logpdf(self, x, c):
        return log(c) + _norm_logpdf(x) + (c - 1) * _norm_logcdf(-x)

    def _cdf(self, x, c):
        return 1.0 - _norm_cdf(-x) ** (c * 1.0)

    def _ppf(self, q, c):
        return -_norm_ppf(pow(1.0 - q, 1.0 / c))
powernorm = powernorm_gen(name='powernorm')


class rdist_gen(rv_continuous):

    """An R-distributed continuous random variable.

    %(before_notes)s

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

        rdist.pdf(x, c) = (1-x**2)**(c/2-1) / B(1/2, c/2)

    for ``-1 <= x <= 1``, ``c > 0``.

    %(example)s

    """

    def _pdf(self, x, c):
        return np.power((1.0 - x ** 2), c / 2.0 - 1) / \
            special.beta(0.5, c / 2.0)

    def _cdf(self, x, c):
        term1 = x / special.beta(0.5, c / 2.0)
        res = 0.5 + term1 * special.hyp2f1(0.5, 1 - c / 2.0, 1.5, x ** 2)
        # There's an issue with hyp2f1, it returns nans near x = +-1, c > 100.
        # Use the generic implementation in that case.  See gh-1285 for
        # background.
        if any(np.isnan(res)):
            return rv_continuous._cdf(self, x, c)
        return res

    def _munp(self, n, c):
        numerator = (1 - (n % 2)) * special.beta((n + 1.0) / 2, c / 2.0)
        return numerator / special.beta(1. / 2, c / 2.)
rdist = rdist_gen(a=-1.0, b=1.0, name="rdist")


class rayleigh_gen(rv_continuous):

    """A Rayleigh continuous random variable.

    %(before_notes)s

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

        rayleigh.pdf(r) = r * exp(-r**2/2)

    for ``x >= 0``.

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

    %(example)s

    """

    def _link(self, x, logSF, phat, ix):
        if ix == 1:
            return x - phat[0] / sqrt(-2.0 * logSF)
        elif ix == 0:
            return x - phat[1] * sqrt(-2.0 * logSF)
        else:
            raise IndexError('Index to the fixed parameter is out of bounds')

    def _rvs(self):
        return chi.rvs(2, size=self._size)

    def _pdf(self, r):
        return exp(self._logpdf(r))

    def _logpdf(self, r):
        rr2 = r * r / 2.0
        return where(rr2 == inf, - rr2, log(r) - rr2)

    def _cdf(self, r):
        return - expm1(-r * r / 2.0)

    def _sf(self, r):
        return exp(-r * r / 2.0)

    def _ppf(self, q):
        return sqrt(-2 * log1p(-q))

    def _isf(self, q):
        return sqrt(-2 * log(q))

    def _stats(self):
        val = 4 - pi
        return (np.sqrt(pi / 2), val / 2, 2 * (pi - 3) * sqrt(pi) / val ** 1.5,
                6 * pi / val - 16 / val ** 2)

    def _entropy(self):
        return _EULER / 2.0 + 1 - 0.5 * log(2)
rayleigh = rayleigh_gen(a=0.0, name="rayleigh")


class truncrayleigh_gen(rv_continuous):

    """A truncated Rayleigh continuous random variable.

    %(before_notes)s

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

        truncrayleigh.cdf(r) = 1 - exp(-((r+c)**2-c**2)/2)

        for ``x >= 0, c>=0``.

    %(example)s

    """

    def _argcheck(self, c):
        return (c >= 0)

    def _link(self, x, logSF, phat, ix):
        c, loc, scale = phat
        if ix == 2:
            return x - loc / (sqrt(c * c - 2 * logSF) - c)
        elif ix == 1:
            return x - scale * (sqrt(c * c - 2 * logSF) - c)
        elif ix == 0:
            xn = (x - loc) / scale
            return - 2 * logSF / xn - xn / 2.0
        else:
            raise IndexError('Index to the fixed parameter is out of bounds')

    def _fitstart(self, data, args=None):
        if args is None:
            args = (0.0,) * self.numargs
        return args + self.fit_loc_scale(data, *args)

    def _pdf(self, r, c):
        rc = r + c
        return rc * exp(-(rc * rc - c * c) / 2.0)

    def _logpdf(self, r, c):
        rc = r + c
        return log(rc) - (rc * rc - c * c) / 2.0

    def _cdf(self, r, c):
        rc = r + c
        return - expm1(-(rc * rc - c * c) / 2.0)

    def _logsf(self, r, c):
        rc = r + c
        return -(rc * rc - c * c) / 2.0

    def _sf(self, r, c):
        return exp(self._logsf(r, c))

    def _ppf(self, q, c):
        return sqrt(c * c - 2 * log1p(-q)) - c

    def _stats(self, c):
        # TODO: correct this it is wrong!
        val = 4 - pi
        return (np.sqrt(pi / 2),
                val / 2,
                2 * (pi - 3) * sqrt(pi) / val ** 1.5,
                6 * pi / val - 16 / val ** 2)

    def _entropy(self, c):
        # TODO: correct this it is wrong!
        return _EULER / 2.0 + 1 - 0.5 * log(2)
truncrayleigh = truncrayleigh_gen(a=0.0, name="truncrayleigh", shapes='c')

# Reciprocal Distribution


class reciprocal_gen(rv_continuous):

    """A reciprocal continuous random variable.

    %(before_notes)s

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

        reciprocal.pdf(x, a, b) = 1 / (x*log(b/a))

    for ``a <= x <= b``, ``a, b > 0``.

    %(example)s

    """

    def _argcheck(self, a, b):
        self.a = a
        self.b = b
        self.d = log(b * 1.0 / a)
        return (a > 0) & (b > 0) & (b > a)

    def _pdf(self, x, a, b):
        return 1.0 / (x * self.d)

    def _logpdf(self, x, a, b):
        return -log(x) - log(self.d)

    def _cdf(self, x, a, b):
        return (log(x) - log(a)) / self.d

    def _ppf(self, q, a, b):
        return a * pow(b * 1.0 / a, q)

    def _munp(self, n, a, b):
        return 1.0 / self.d / n * (pow(b * 1.0, n) - pow(a * 1.0, n))

    def _fitstart(self, data):
        a = np.min(data)
        a -= 0.01 * np.abs(a)
        b = np.max(data)
        b += 0.01 * np.abs(b)
        if a <= 0:
            da = np.abs(a) + 0.001
            a += da
            b += da
        return super(reciprocal_gen, self)._fitstart(data, args=(a, b))

    def _entropy(self, a, b):
        return 0.5 * log(a * b) + log(log(b / a))
reciprocal = reciprocal_gen(name="reciprocal")


# FIXME: PPF does not work.
class rice_gen(rv_continuous):

    """A Rice continuous random variable.

    %(before_notes)s

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

        rice.pdf(x, b) = x * exp(-(x**2+b**2)/2) * I[0](x*b)

    for ``x > 0``, ``b > 0``.

    %(example)s

    """

    def _argcheck(self, b):
        return b >= 0

    def _rvs(self, b):
        # http://en.wikipedia.org/wiki/Rice_distribution
        sz = self._size if self._size else 1
        t = b / np.sqrt(2) + mtrand.standard_normal(size=(2, sz))
        return np.sqrt((t * t).sum(axis=0))

    def _pdf(self, x, b):
        return x * exp(-(x - b) * (x - b) / 2.0) * special.i0e(x * b)

    def _munp(self, n, b):
        nd2 = n / 2.0
        n1 = 1 + nd2
        b2 = b * b / 2.0
        return (2.0 ** (nd2) * exp(-b2) * special.gamma(n1) *
                special.hyp1f1(n1, 1, b2))
rice = rice_gen(a=0.0, name="rice")


# FIXME: PPF does not work.
class recipinvgauss_gen(rv_continuous):

    """A reciprocal inverse Gaussian continuous random variable.

    %(before_notes)s

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

        recipinvgauss.pdf(x, mu) = 1/sqrt(2*pi*x) * exp(-(1-mu*x)**2/(2*x*mu**2))

    for ``x >= 0``.

    %(example)s

    """

    def _rvs(self, mu):
        return 1.0 / mtrand.wald(mu, 1.0, size=self._size)

    def _pdf(self, x, mu):
        return 1.0 / sqrt(2 * pi * x) * \
            exp(-(1 - mu * x) ** 2.0 / (2 * x * mu ** 2.0))

    def _logpdf(self, x, mu):
        return -(1 - mu * x) ** 2.0 / \
            (2 * x * mu ** 2.0) - 0.5 * log(2 * pi * x)

    def _cdf(self, x, mu):
        trm1 = 1.0 / mu - x
        trm2 = 1.0 / mu + x
        isqx = 1.0 / sqrt(x)
        return 1.0 - _norm_cdf(isqx * trm1) - \
            exp(2.0 / mu) * _norm_cdf(-isqx * trm2)
recipinvgauss = recipinvgauss_gen(a=0.0, name='recipinvgauss')


class semicircular_gen(rv_continuous):

    """A semicircular continuous random variable.

    %(before_notes)s

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

        semicircular.pdf(x) = 2/pi * sqrt(1-x**2)

    for ``-1 <= x <= 1``.

    %(example)s

    """

    def _pdf(self, x):
        return 2.0 / pi * sqrt(1 - x * x)

    def _cdf(self, x):
        return 0.5 + 1.0 / pi * (x * sqrt(1 - x * x) + arcsin(x))

    def _stats(self):
        return 0, 0.25, 0, -1.0

    def _entropy(self):
        return 0.64472988584940017414
semicircular = semicircular_gen(a=-1.0, b=1.0, name="semicircular")


class triang_gen(rv_continuous):

    """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)``.

    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

    """

    def _rvs(self, c):
        return mtrand.triangular(0, c, 1, self._size)

    def _argcheck(self, c):
        return (c >= 0) & (c <= 1)

    def _pdf(self, x, c):
        return where(x < c, 2 * x / c, 2 * (1 - x) / (1 - c))

    def _cdf(self, x, c):
        return where(x < c, x * x / c, (x * x - 2 * x + c) / (c - 1))

    def _ppf(self, q, c):
        return where(q < c, sqrt(c * q), 1 - sqrt((1 - c) * (1 - q)))

    def _stats(self, c):
        return (c + 1.0) / 3.0, (1.0 - c + c * c) / 18, sqrt(2) * (2 * c - 1) * (c + 1) * (c - 2) / \
               (5 * np.power((1.0 - c + c * c), 1.5)), -3.0 / 5.0

    def _entropy(self, c):
        return 0.5 - log(2)
triang = triang_gen(a=0.0, b=1.0, name="triang")


class truncexpon_gen(rv_continuous):

    """A truncated exponential continuous random variable.

    %(before_notes)s

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

        truncexpon.pdf(x, b) = exp(-x) / (1-exp(-b))

    for ``0 < x < b``.

    %(example)s

    """

    def _argcheck(self, b):
        self.b = b
        return (b > 0)

    def _pdf(self, x, b):
        return exp(-x) / (-expm1(-b))

    def _logpdf(self, x, b):
        return - x - log(-expm1(-b))

    def _cdf(self, x, b):
        return expm1(-x) / expm1(-b)

    def _ppf(self, q, b):
        return - log1p(q * expm1(-b))

    def _munp(self, n, b):
        # wrong answer with formula, same as in continuous.pdf
        # return gam(n+1)-special.gammainc(1+n, b)
        if n == 1:
            return (1 - (b + 1) * exp(-b)) / (-expm1(-b))
        elif n == 2:
            return 2 * (1 - 0.5 * (b * b + 2 * b + 2) * exp(-b)) / (-expm1(-b))
        else:
            # return generic for higher moments
            # return rv_continuous._mom1_sc(self, n, b)
            return self._mom1_sc(n, b)

    def _entropy(self, b):
        eB = exp(b)
        return log(eB - 1) + (1 + eB * (b - 1.0)) / (1.0 - eB)
truncexpon = truncexpon_gen(a=0.0, name='truncexpon')


class truncnorm_gen(rv_continuous):

    """A truncated normal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The standard form of this distribution is a standard normal truncated to
    the range [a, b] --- notice that a and b are defined over the domain of the
    standard normal.  To convert clip values for a specific mean and standard
    deviation, use::

        a, b = (myclip_a - my_mean) / my_std, (myclip_b - my_mean) / my_std

    %(example)s

    """

    def _argcheck(self, a, b):
        self.a = a
        self.b = b
        self._nb = _norm_cdf(b)
        self._na = _norm_cdf(a)
        self._sb = _norm_sf(b)
        self._sa = _norm_sf(a)
        if self.a > 0:
            self._delta = -(self._sb - self._sa)
        else:
            self._delta = self._nb - self._na
        self._logdelta = log(self._delta)
        return (a != b)

    def _pdf(self, x, a, b):
        return _norm_pdf(x) / self._delta

    def _logpdf(self, x, a, b):
        return _norm_logpdf(x) - self._logdelta

    def _cdf(self, x, a, b):
        return (_norm_cdf(x) - self._na) / self._delta

    def _ppf(self, q, a, b):
        if self.a > 0:
            return _norm_isf(q * self._sb + self._sa * (1.0 - q))
        else:
            return _norm_ppf(q * self._nb + self._na * (1.0 - q))

    def _stats(self, a, b):
        nA, nB = self._na, self._nb
        d = nB - nA
        pA, pB = _norm_pdf(a), _norm_pdf(b)
        mu = (pA - pB) / d   # correction sign
        mu2 = 1 + (a * pA - b * pB) / d - mu * mu
        return mu, mu2, None, None
truncnorm = truncnorm_gen(name='truncnorm')


# FIXME: RVS does not work.
class tukeylambda_gen(rv_continuous):

    """A Tukey-Lamdba continuous random variable.

    %(before_notes)s

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

        - Cauchy                (lam=-1)
        - logistic              (lam=0.0)
        - approx Normal         (lam=0.14)
        - u-shape               (lam = 0.5)
        - uniform from -1 to 1  (lam = 1)

    %(example)s

    """

    def _argcheck(self, lam):
        return np.ones(np.shape(lam), dtype=bool)

    def _pdf(self, x, lam):
        Fx = asarray(special.tklmbda(x, lam))
        Px = Fx ** (lam - 1.0) + (asarray(1 - Fx)) ** (lam - 1.0)
        Px = 1.0 / asarray(Px)
        return where((lam <= 0) | (abs(x) < 1.0 / asarray(lam)), Px, 0.0)

    def _cdf(self, x, lam):
        return special.tklmbda(x, lam)

    def _ppf(self, q, lam):
        q = q * 1.0
        vals1 = (q ** lam - (1 - q) ** lam) / lam
        vals2 = log(q / (1 - q))
        return where((lam == 0) & (q == q), vals2, vals1)

    def _stats(self, lam):
        return 0, _tlvar(lam), 0, _tlkurt(lam)

    def _entropy(self, lam):
        def integ(p):
            return log(pow(p, lam - 1) + pow(1 - p, lam - 1))
        return integrate.quad(integ, 0, 1)[0]
tukeylambda = tukeylambda_gen(name='tukeylambda')


class uniform_gen(rv_continuous):

    """A uniform continuous random variable.

    This distribution is constant between `loc` and ``loc + scale``.

    %(before_notes)s

    %(example)s

    """

    def _rvs(self):
        return mtrand.uniform(0.0, 1.0, self._size)

    def _pdf(self, x):
        return 1.0 * (x == x)

    def _cdf(self, x):
        return x

    def _ppf(self, q):
        return q

    def _stats(self):
        return 0.5, 1.0 / 12, 0, -1.2

    def _entropy(self):
        return 0.0
uniform = uniform_gen(a=0.0, b=1.0, name='uniform')


class vonmises_gen(rv_continuous):

    """A Von Mises continuous random variable.

    %(before_notes)s

    Notes
    -----
    If `x` is not in range or `loc` is not in range it assumes they are angles
    and converts them to [-pi, pi] equivalents.

    The probability density function for `vonmises` is::

        vonmises.pdf(x, kappa) = exp(kappa * cos(x)) / (2*pi*I[0](kappa))

    for ``-pi <= x <= pi``, ``kappa > 0``.

    See Also
    --------
    vonmises_line : The same distribution, defined on a [-pi, pi] segment
                    of the real line.

    %(example)s

    """

    def _rvs(self, kappa):
        return mtrand.vonmises(0.0, kappa, size=self._size)

    def _pdf(self, x, kappa):
        return exp(kappa * cos(x)) / (2 * pi * special.i0(kappa))

    def _cdf(self, x, kappa):
        return vonmises_cython.von_mises_cdf(kappa, x)

    def _stats_skip(self, kappa):
        return 0, None, 0, None
vonmises = vonmises_gen(name='vonmises')
vonmises_line = vonmises_gen(a=-np.pi, b=np.pi, name='vonmises_line')


class wald_gen(invgauss_gen):

    """A Wald continuous random variable.

    %(before_notes)s

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

        wald.pdf(x) = 1/sqrt(2*pi*x**3) * exp(-(x-1)**2/(2*x))

    for ``x > 0``.

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

    %(example)s
    """

    def _rvs(self):
        return mtrand.wald(1.0, 1.0, size=self._size)

    def _pdf(self, x):
        return invgauss._pdf(x, 1.0)

    def _logpdf(self, x):
        return invgauss._logpdf(x, 1.0)

    def _cdf(self, x):
        return invgauss._cdf(x, 1.0)

    def _stats(self):
        return 1.0, 1.0, 3.0, 15.0
wald = wald_gen(a=0.0, name="wald")


class wrapcauchy_gen(rv_continuous):

    """A wrapped Cauchy continuous random variable.

    %(before_notes)s

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

        wrapcauchy.pdf(x, c) = (1-c**2) / (2*pi*(1+c**2-2*c*cos(x)))

    for ``0 <= x <= 2*pi``, ``0 < c < 1``.

    %(example)s

    """

    def _argcheck(self, c):
        return (c > 0) & (c < 1)

    def _pdf(self, x, c):
        return (1.0 - c * c) / (2 * pi * (1 + c * c - 2 * c * cos(x)))

    def _cdf(self, x, c):
        output = 0.0 * x
        val = (1.0 + c) / (1.0 - c)
        c1 = x < pi
        c2 = 1 - c1
        xp = extract(c1, x)
        xn = extract(c2, x)
        if (any(xn)):
            valn = extract(c2, np.ones_like(x) * val)
            xn = 2 * pi - xn
            yn = tan(xn / 2.0)
            on = 1.0 - 1.0 / pi * arctan(valn * yn)
            place(output, c2, on)
        if (any(xp)):
            valp = extract(c1, np.ones_like(x) * val)
            yp = tan(xp / 2.0)
            op = 1.0 / pi * arctan(valp * yp)
            place(output, c1, op)
        return output

    def _ppf(self, q, c):
        val = (1.0 - c) / (1.0 + c)
        rcq = 2 * arctan(val * tan(pi * q))
        rcmq = 2 * pi - 2 * arctan(val * tan(pi * (1 - q)))
        return where(q < 1.0 / 2, rcq, rcmq)

    def _entropy(self, c):
        return log(2 * pi * (1 - c * c))
wrapcauchy = wrapcauchy_gen(a=0.0, b=2 * pi, name='wrapcauchy')


# Collect names of classes and objects in this module.
pairs = list(globals().items())
_distn_names, _distn_gen_names = get_distribution_names(pairs, rv_continuous)

__all__ = _distn_names + _distn_gen_names