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### boost/math/distributions/skew_normal.hpp

```//  Copyright Benjamin Sobotta 2012

//  Use, modification and distribution are subject to the
//  Boost Software License, Version 1.0. (See accompanying file

#ifndef BOOST_STATS_SKEW_NORMAL_HPP
#define BOOST_STATS_SKEW_NORMAL_HPP

// http://en.wikipedia.org/wiki/Skew_normal_distribution
// http://azzalini.stat.unipd.it/SN/
// Also:
// Azzalini, A. (1985). "A class of distributions which includes the normal ones".
// Scand. J. Statist. 12: 171-178.

#include <boost/math/distributions/fwd.hpp> // TODO add skew_normal distribution to fwd.hpp!
#include <boost/math/special_functions/owens_t.hpp> // Owen's T function
#include <boost/math/distributions/complement.hpp>
#include <boost/math/distributions/normal.hpp>
#include <boost/math/distributions/detail/common_error_handling.hpp>
#include <boost/math/constants/constants.hpp>
#include <boost/math/tools/tuple.hpp>
#include <boost/math/tools/roots.hpp> // Newton-Raphson
#include <boost/math/tools/assert.hpp>
#include <boost/math/distributions/detail/generic_mode.hpp> // pdf max finder.

#include <utility>
#include <algorithm> // std::lower_bound, std::distance

namespace boost{ namespace math{

namespace detail
{
template <class RealType, class Policy>
inline bool check_skew_normal_shape(
const char* function,
RealType shape,
RealType* result,
const Policy& pol)
{
if(!(boost::math::isfinite)(shape))
{
*result =
policies::raise_domain_error<RealType>(function,
"Shape parameter is %1%, but must be finite!",
shape, pol);
return false;
}
return true;
}

} // namespace detail

template <class RealType = double, class Policy = policies::policy<> >
class skew_normal_distribution
{
public:
typedef RealType value_type;
typedef Policy policy_type;

skew_normal_distribution(RealType l_location = 0, RealType l_scale = 1, RealType l_shape = 0)
: location_(l_location), scale_(l_scale), shape_(l_shape)
{ // Default is a 'standard' normal distribution N01. (shape=0 results in the normal distribution with no skew)
static const char* function = "boost::math::skew_normal_distribution<%1%>::skew_normal_distribution";

RealType result;
detail::check_scale(function, l_scale, &result, Policy());
detail::check_location(function, l_location, &result, Policy());
detail::check_skew_normal_shape(function, l_shape, &result, Policy());
}

RealType location()const
{
return location_;
}

RealType scale()const
{
return scale_;
}

RealType shape()const
{
return shape_;
}

private:
//
// Data members:
//
RealType location_;  // distribution location.
RealType scale_;    // distribution scale.
RealType shape_;    // distribution shape.
}; // class skew_normal_distribution

typedef skew_normal_distribution<double> skew_normal;

#ifdef __cpp_deduction_guides
template <class RealType>
skew_normal_distribution(RealType)->skew_normal_distribution<typename boost::math::tools::promote_args<RealType>::type>;
template <class RealType>
skew_normal_distribution(RealType,RealType)->skew_normal_distribution<typename boost::math::tools::promote_args<RealType>::type>;
template <class RealType>
skew_normal_distribution(RealType,RealType,RealType)->skew_normal_distribution<typename boost::math::tools::promote_args<RealType>::type>;
#endif

template <class RealType, class Policy>
inline const std::pair<RealType, RealType> range(const skew_normal_distribution<RealType, Policy>& /*dist*/)
{ // Range of permissible values for random variable x.
using boost::math::tools::max_value;
return std::pair<RealType, RealType>(
std::numeric_limits<RealType>::has_infinity ? -std::numeric_limits<RealType>::infinity() : -max_value<RealType>(),
std::numeric_limits<RealType>::has_infinity ? std::numeric_limits<RealType>::infinity() : max_value<RealType>()); // - to + max value.
}

template <class RealType, class Policy>
inline const std::pair<RealType, RealType> support(const skew_normal_distribution<RealType, Policy>& /*dist*/)
{ // Range of supported values for random variable x.
// This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.

using boost::math::tools::max_value;
return std::pair<RealType, RealType>(-max_value<RealType>(),  max_value<RealType>()); // - to + max value.
}

template <class RealType, class Policy>
inline RealType pdf(const skew_normal_distribution<RealType, Policy>& dist, const RealType& x)
{
const RealType scale = dist.scale();
const RealType location = dist.location();
const RealType shape = dist.shape();

static const char* function = "boost::math::pdf(const skew_normal_distribution<%1%>&, %1%)";

RealType result = 0;
if(false == detail::check_scale(function, scale, &result, Policy()))
{
return result;
}
if(false == detail::check_location(function, location, &result, Policy()))
{
return result;
}
if(false == detail::check_skew_normal_shape(function, shape, &result, Policy()))
{
return result;
}
if((boost::math::isinf)(x))
{
return 0; // pdf + and - infinity is zero.
}
// Below produces MSVC 4127 warnings, so the above used instead.
//if(std::numeric_limits<RealType>::has_infinity && abs(x) == std::numeric_limits<RealType>::infinity())
//{ // pdf + and - infinity is zero.
//  return 0;
//}
if(false == detail::check_x(function, x, &result, Policy()))
{
return result;
}

const RealType transformed_x = (x-location)/scale;

normal_distribution<RealType, Policy> std_normal;

result = pdf(std_normal, transformed_x) * cdf(std_normal, shape*transformed_x) * 2 / scale;

return result;
} // pdf

template <class RealType, class Policy>
inline RealType cdf(const skew_normal_distribution<RealType, Policy>& dist, const RealType& x)
{
const RealType scale = dist.scale();
const RealType location = dist.location();
const RealType shape = dist.shape();

static const char* function = "boost::math::cdf(const skew_normal_distribution<%1%>&, %1%)";
RealType result = 0;
if(false == detail::check_scale(function, scale, &result, Policy()))
{
return result;
}
if(false == detail::check_location(function, location, &result, Policy()))
{
return result;
}
if(false == detail::check_skew_normal_shape(function, shape, &result, Policy()))
{
return result;
}
if((boost::math::isinf)(x))
{
if(x < 0) return 0; // -infinity
return 1; // + infinity
}
// These produce MSVC 4127 warnings, so the above used instead.
//if(std::numeric_limits<RealType>::has_infinity && x == std::numeric_limits<RealType>::infinity())
//{ // cdf +infinity is unity.
//  return 1;
//}
//if(std::numeric_limits<RealType>::has_infinity && x == -std::numeric_limits<RealType>::infinity())
//{ // cdf -infinity is zero.
//  return 0;
//}
if(false == detail::check_x(function, x, &result, Policy()))
{
return result;
}

const RealType transformed_x = (x-location)/scale;

normal_distribution<RealType, Policy> std_normal;

result = cdf(std_normal, transformed_x) - owens_t(transformed_x, shape)*static_cast<RealType>(2);

return result;
} // cdf

template <class RealType, class Policy>
inline RealType cdf(const complemented2_type<skew_normal_distribution<RealType, Policy>, RealType>& c)
{
const RealType scale = c.dist.scale();
const RealType location = c.dist.location();
const RealType shape = c.dist.shape();
const RealType x = c.param;

static const char* function = "boost::math::cdf(const complement(skew_normal_distribution<%1%>&), %1%)";

if((boost::math::isinf)(x))
{
if(x < 0) return 1; // cdf complement -infinity is unity.
return 0; // cdf complement +infinity is zero
}
// These produce MSVC 4127 warnings, so the above used instead.
//if(std::numeric_limits<RealType>::has_infinity && x == std::numeric_limits<RealType>::infinity())
//{ // cdf complement +infinity is zero.
//  return 0;
//}
//if(std::numeric_limits<RealType>::has_infinity && x == -std::numeric_limits<RealType>::infinity())
//{ // cdf complement -infinity is unity.
//  return 1;
//}
RealType result = 0;
if(false == detail::check_scale(function, scale, &result, Policy()))
return result;
if(false == detail::check_location(function, location, &result, Policy()))
return result;
if(false == detail::check_skew_normal_shape(function, shape, &result, Policy()))
return result;
if(false == detail::check_x(function, x, &result, Policy()))
return result;

const RealType transformed_x = (x-location)/scale;

normal_distribution<RealType, Policy> std_normal;

result = cdf(complement(std_normal, transformed_x)) + owens_t(transformed_x, shape)*static_cast<RealType>(2);
return result;
} // cdf complement

template <class RealType, class Policy>
inline RealType location(const skew_normal_distribution<RealType, Policy>& dist)
{
return dist.location();
}

template <class RealType, class Policy>
inline RealType scale(const skew_normal_distribution<RealType, Policy>& dist)
{
return dist.scale();
}

template <class RealType, class Policy>
inline RealType shape(const skew_normal_distribution<RealType, Policy>& dist)
{
return dist.shape();
}

template <class RealType, class Policy>
inline RealType mean(const skew_normal_distribution<RealType, Policy>& dist)
{
BOOST_MATH_STD_USING  // for ADL of std functions

using namespace boost::math::constants;

//const RealType delta = dist.shape() / sqrt(static_cast<RealType>(1)+dist.shape()*dist.shape());

//return dist.location() + dist.scale() * delta * root_two_div_pi<RealType>();

return dist.location() + dist.scale() * dist.shape() / sqrt(pi<RealType>()+pi<RealType>()*dist.shape()*dist.shape()) * root_two<RealType>();
}

template <class RealType, class Policy>
inline RealType variance(const skew_normal_distribution<RealType, Policy>& dist)
{
using namespace boost::math::constants;

const RealType delta2 = dist.shape() != 0 ? static_cast<RealType>(1) / (static_cast<RealType>(1)+static_cast<RealType>(1)/(dist.shape()*dist.shape())) : static_cast<RealType>(0);
//const RealType inv_delta2 = static_cast<RealType>(1)+static_cast<RealType>(1)/(dist.shape()*dist.shape());

RealType variance = dist.scale()*dist.scale()*(static_cast<RealType>(1)-two_div_pi<RealType>()*delta2);
//RealType variance = dist.scale()*dist.scale()*(static_cast<RealType>(1)-two_div_pi<RealType>()/inv_delta2);

return variance;
}

namespace detail
{
/*
TODO No closed expression for mode, so use max of pdf.
*/

template <class RealType, class Policy>
inline RealType mode_fallback(const skew_normal_distribution<RealType, Policy>& dist)
{ // mode.
static const char* function = "mode(skew_normal_distribution<%1%> const&)";
const RealType scale = dist.scale();
const RealType location = dist.location();
const RealType shape = dist.shape();

RealType result;
if(!detail::check_scale(
function,
scale, &result, Policy())
||
!detail::check_skew_normal_shape(
function,
shape,
&result,
Policy()))
return result;

if( shape == 0 )
{
return location;
}

if( shape < 0 )
{
skew_normal_distribution<RealType, Policy> D(0, 1, -shape);
result = mode_fallback(D);
result = location-scale*result;
return result;
}

BOOST_MATH_STD_USING

// 21 elements
static const RealType shapes[] = {
0.0,
1.000000000000000e-004,
2.069138081114790e-004,
4.281332398719396e-004,
8.858667904100824e-004,
1.832980710832436e-003,
3.792690190732250e-003,
7.847599703514606e-003,
1.623776739188722e-002,
3.359818286283781e-002,
6.951927961775606e-002,
1.438449888287663e-001,
2.976351441631319e-001,
6.158482110660261e-001,
1.274274985703135e+000,
2.636650898730361e+000,
5.455594781168514e+000,
1.128837891684688e+001,
2.335721469090121e+001,
4.832930238571753e+001,
1.000000000000000e+002};

// 21 elements
static const RealType guess[] = {
0.0,
5.000050000525391e-005,
1.500015000148736e-004,
3.500035000350010e-004,
7.500075000752560e-004,
1.450014500145258e-003,
3.050030500305390e-003,
6.250062500624765e-003,
1.295012950129504e-002,
2.675026750267495e-002,
5.525055250552491e-002,
1.132511325113255e-001,
2.249522495224952e-001,
3.992539925399257e-001,
5.353553535535358e-001,
4.954549545495457e-001,
3.524535245352451e-001,
2.182521825218249e-001,
1.256512565125654e-001,
6.945069450694508e-002,
3.735037350373460e-002
};

const RealType* result_ptr = std::lower_bound(shapes, shapes+21, shape);

typedef typename std::iterator_traits<RealType*>::difference_type diff_type;

const diff_type d = std::distance(shapes, result_ptr);

BOOST_MATH_ASSERT(d > static_cast<diff_type>(0));

// refine
if(d < static_cast<diff_type>(21)) // shape smaller 100
{
result = guess[d-static_cast<diff_type>(1)]
+ (guess[d]-guess[d-static_cast<diff_type>(1)])/(shapes[d]-shapes[d-static_cast<diff_type>(1)])
* (shape-shapes[d-static_cast<diff_type>(1)]);
}
else // shape greater 100
{
result = 1e-4;
}

skew_normal_distribution<RealType, Policy> helper(0, 1, shape);

result = detail::generic_find_mode_01(helper, result, function);

result = result*scale + location;

return result;
} // mode_fallback

/*
* TODO No closed expression for mode, so use f'(x) = 0
*/
template <class RealType, class Policy>
struct skew_normal_mode_functor
{
skew_normal_mode_functor(const boost::math::skew_normal_distribution<RealType, Policy> dist)
: distribution(dist)
{
}

boost::math::tuple<RealType, RealType> operator()(RealType const& x)
{
normal_distribution<RealType, Policy> std_normal;
const RealType shape = distribution.shape();
const RealType pdf_x = pdf(distribution, x);
const RealType normpdf_x = pdf(std_normal, x);
const RealType normpdf_ax = pdf(std_normal, x*shape);
RealType fx = static_cast<RealType>(2)*shape*normpdf_ax*normpdf_x - x*pdf_x;
RealType dx = static_cast<RealType>(2)*shape*x*normpdf_x*normpdf_ax*(static_cast<RealType>(1) + shape*shape) + pdf_x + x*fx;
// return both function evaluation difference f(x) and 1st derivative f'(x).
return boost::math::make_tuple(fx, -dx);
}
private:
const boost::math::skew_normal_distribution<RealType, Policy> distribution;
};

} // namespace detail

template <class RealType, class Policy>
inline RealType mode(const skew_normal_distribution<RealType, Policy>& dist)
{
const RealType scale = dist.scale();
const RealType location = dist.location();
const RealType shape = dist.shape();

static const char* function = "boost::math::mode(const skew_normal_distribution<%1%>&, %1%)";

RealType result = 0;
if(false == detail::check_scale(function, scale, &result, Policy()))
return result;
if(false == detail::check_location(function, location, &result, Policy()))
return result;
if(false == detail::check_skew_normal_shape(function, shape, &result, Policy()))
return result;

if( shape == 0 )
{
return location;
}

if( shape < 0 )
{
skew_normal_distribution<RealType, Policy> D(0, 1, -shape);
result = mode(D);
result = location-scale*result;
return result;
}

// 21 elements
static const RealType shapes[] = {
static_cast<RealType>(0.0),
static_cast<RealType>(1.000000000000000e-004),
static_cast<RealType>(2.069138081114790e-004),
static_cast<RealType>(4.281332398719396e-004),
static_cast<RealType>(8.858667904100824e-004),
static_cast<RealType>(1.832980710832436e-003),
static_cast<RealType>(3.792690190732250e-003),
static_cast<RealType>(7.847599703514606e-003),
static_cast<RealType>(1.623776739188722e-002),
static_cast<RealType>(3.359818286283781e-002),
static_cast<RealType>(6.951927961775606e-002),
static_cast<RealType>(1.438449888287663e-001),
static_cast<RealType>(2.976351441631319e-001),
static_cast<RealType>(6.158482110660261e-001),
static_cast<RealType>(1.274274985703135e+000),
static_cast<RealType>(2.636650898730361e+000),
static_cast<RealType>(5.455594781168514e+000),
static_cast<RealType>(1.128837891684688e+001),
static_cast<RealType>(2.335721469090121e+001),
static_cast<RealType>(4.832930238571753e+001),
static_cast<RealType>(1.000000000000000e+002)
};

// 21 elements
static const RealType guess[] = {
static_cast<RealType>(0.0),
static_cast<RealType>(5.000050000525391e-005),
static_cast<RealType>(1.500015000148736e-004),
static_cast<RealType>(3.500035000350010e-004),
static_cast<RealType>(7.500075000752560e-004),
static_cast<RealType>(1.450014500145258e-003),
static_cast<RealType>(3.050030500305390e-003),
static_cast<RealType>(6.250062500624765e-003),
static_cast<RealType>(1.295012950129504e-002),
static_cast<RealType>(2.675026750267495e-002),
static_cast<RealType>(5.525055250552491e-002),
static_cast<RealType>(1.132511325113255e-001),
static_cast<RealType>(2.249522495224952e-001),
static_cast<RealType>(3.992539925399257e-001),
static_cast<RealType>(5.353553535535358e-001),
static_cast<RealType>(4.954549545495457e-001),
static_cast<RealType>(3.524535245352451e-001),
static_cast<RealType>(2.182521825218249e-001),
static_cast<RealType>(1.256512565125654e-001),
static_cast<RealType>(6.945069450694508e-002),
static_cast<RealType>(3.735037350373460e-002)
};

const RealType* result_ptr = std::lower_bound(shapes, shapes+21, shape);

typedef typename std::iterator_traits<RealType*>::difference_type diff_type;

const diff_type d = std::distance(shapes, result_ptr);

BOOST_MATH_ASSERT(d > static_cast<diff_type>(0));

// TODO: make the search bounds smarter, depending on the shape parameter
RealType search_min = 0; // below zero was caught above
RealType search_max = 0.55f; // will never go above 0.55

// refine
if(d < static_cast<diff_type>(21)) // shape smaller 100
{
// it is safe to assume that d > 0, because shape==0.0 is caught earlier
result = guess[d-static_cast<diff_type>(1)]
+ (guess[d]-guess[d-static_cast<diff_type>(1)])/(shapes[d]-shapes[d-static_cast<diff_type>(1)])
* (shape-shapes[d-static_cast<diff_type>(1)]);
}
else // shape greater 100
{
result = 1e-4f;
search_max = guess[19]; // set 19 instead of 20 to have a safety margin because the table may not be exact @ shape=100
}

const int get_digits = policies::digits<RealType, Policy>();// get digits from policy,
std::uintmax_t max_iter = policies::get_max_root_iterations<Policy>(); // and max iterations.

skew_normal_distribution<RealType, Policy> helper(0, 1, shape);

result = tools::newton_raphson_iterate(detail::skew_normal_mode_functor<RealType, Policy>(helper), result,
search_min, search_max, get_digits, max_iter);
if (max_iter >= policies::get_max_root_iterations<Policy>())
{
return policies::raise_evaluation_error<RealType>(function, "Unable to locate solution in a reasonable time:" // LCOV_EXCL_LINE
" either there is no answer to quantile or the answer is infinite.  Current best guess is %1%", result, Policy()); // LCOV_EXCL_LINE
}

result = result*scale + location;

return result;
}

template <class RealType, class Policy>
inline RealType skewness(const skew_normal_distribution<RealType, Policy>& dist)
{
BOOST_MATH_STD_USING  // for ADL of std functions
using namespace boost::math::constants;

static const RealType factor = four_minus_pi<RealType>()/static_cast<RealType>(2);
const RealType delta = dist.shape() / sqrt(static_cast<RealType>(1)+dist.shape()*dist.shape());

return static_cast<RealType>(factor * pow(root_two_div_pi<RealType>() * delta, 3) /
pow(static_cast<RealType>(1)-two_div_pi<RealType>()*delta*delta, static_cast<RealType>(1.5)));
}

template <class RealType, class Policy>
inline RealType kurtosis(const skew_normal_distribution<RealType, Policy>& dist)
{
return kurtosis_excess(dist)+static_cast<RealType>(3);
}

template <class RealType, class Policy>
inline RealType kurtosis_excess(const skew_normal_distribution<RealType, Policy>& dist)
{
using namespace boost::math::constants;

static const RealType factor = pi_minus_three<RealType>()*static_cast<RealType>(2);

const RealType delta2 = dist.shape() != 0 ? static_cast<RealType>(1) / (static_cast<RealType>(1)+static_cast<RealType>(1)/(dist.shape()*dist.shape())) : static_cast<RealType>(0);

const RealType x = static_cast<RealType>(1)-two_div_pi<RealType>()*delta2;
const RealType y = two_div_pi<RealType>() * delta2;

return factor * y*y / (x*x);
}

namespace detail
{

template <class RealType, class Policy>
struct skew_normal_quantile_functor
{
skew_normal_quantile_functor(const boost::math::skew_normal_distribution<RealType, Policy> dist, RealType const& p)
: distribution(dist), prob(p)
{
}

boost::math::tuple<RealType, RealType> operator()(RealType const& x)
{
RealType c = cdf(distribution, x);
RealType fx = c - prob;  // Difference cdf - value - to minimize.
RealType dx = pdf(distribution, x); // pdf is 1st derivative.
// return both function evaluation difference f(x) and 1st derivative f'(x).
return boost::math::make_tuple(fx, dx);
}
private:
const boost::math::skew_normal_distribution<RealType, Policy> distribution;
RealType prob;
};

} // namespace detail

template <class RealType, class Policy>
inline RealType quantile(const skew_normal_distribution<RealType, Policy>& dist, const RealType& p)
{
const RealType scale = dist.scale();
const RealType location = dist.location();
const RealType shape = dist.shape();

static const char* function = "boost::math::quantile(const skew_normal_distribution<%1%>&, %1%)";

RealType result = 0;
if(false == detail::check_scale(function, scale, &result, Policy()))
return result;
if(false == detail::check_location(function, location, &result, Policy()))
return result;
if(false == detail::check_skew_normal_shape(function, shape, &result, Policy()))
return result;
if(false == detail::check_probability(function, p, &result, Policy()))
return result;

// Compute initial guess via Cornish-Fisher expansion.
RealType x = -boost::math::erfc_inv(2 * p, Policy()) * constants::root_two<RealType>();

// Avoid unnecessary computations if there is no skew.
if(shape != 0)
{
const RealType skew = skewness(dist);
const RealType exk = kurtosis_excess(dist);

x = x + (x*x-static_cast<RealType>(1))*skew/static_cast<RealType>(6)
+ x*(x*x-static_cast<RealType>(3))*exk/static_cast<RealType>(24)
- x*(static_cast<RealType>(2)*x*x-static_cast<RealType>(5))*skew*skew/static_cast<RealType>(36);
} // if(shape != 0)

result = standard_deviation(dist)*x+mean(dist);

// handle special case of non-skew normal distribution.
if(shape == 0)
return result;

// refine the result by numerically searching the root of (p-cdf)

const RealType search_min = support(dist).first;
const RealType search_max = support(dist).second;

const int get_digits = policies::digits<RealType, Policy>();// get digits from policy,
std::uintmax_t max_iter = policies::get_max_root_iterations<Policy>(); // and max iterations.

result = tools::newton_raphson_iterate(detail::skew_normal_quantile_functor<RealType, Policy>(dist, p), result,
search_min, search_max, get_digits, max_iter);
if (max_iter >= policies::get_max_root_iterations<Policy>())
{
return policies::raise_evaluation_error<RealType>(function, "Unable to locate solution in a reasonable time: either there is no answer to quantile" // LCOV_EXCL_LINE
" or the answer is infinite.  Current best guess is %1%", result, Policy());  // LCOV_EXCL_LINE
}

return result;
} // quantile

template <class RealType, class Policy>
inline RealType quantile(const complemented2_type<skew_normal_distribution<RealType, Policy>, RealType>& c)
{
const RealType scale = c.dist.scale();
const RealType location = c.dist.location();
const RealType shape = c.dist.shape();

static const char* function = "boost::math::quantile(const complement(skew_normal_distribution<%1%>&), %1%)";
RealType result = 0;
if(false == detail::check_scale(function, scale, &result, Policy()))
return result;
if(false == detail::check_location(function, location, &result, Policy()))
return result;
if(false == detail::check_skew_normal_shape(function, shape, &result, Policy()))
return result;
RealType q = c.param;
if(false == detail::check_probability(function, q, &result, Policy()))
return result;

skew_normal_distribution<RealType, Policy> D(-location, scale, -shape);

result = -quantile(D, q);

return result;
} // quantile

} // namespace math
} // namespace boost

// This include must be at the end, *after* the accessors
// for this distribution have been defined, in order to
// keep compilers that support two-phase lookup happy.
#include <boost/math/distributions/detail/derived_accessors.hpp>

#endif // BOOST_STATS_SKEW_NORMAL_HPP

```