// Copyright John Maddock 2006.
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// Copyright Paul A. Bristow 2006, 2012, 2017.
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// Copyright Thomas Mang 2012.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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#ifndef BOOST_STATS_STUDENTS_T_HPP
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#define BOOST_STATS_STUDENTS_T_HPP
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// http://en.wikipedia.org/wiki/Student%27s_t_distribution
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// http://www.itl.nist.gov/div898/handbook/eda/section3/eda3664.htm
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#include <boost/math/distributions/fwd.hpp>
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#include <boost/math/special_functions/beta.hpp> // for ibeta(a, b, x).
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#include <boost/math/special_functions/digamma.hpp>
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#include <boost/math/distributions/complement.hpp>
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#include <boost/math/distributions/detail/common_error_handling.hpp>
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#include <boost/math/distributions/normal.hpp>
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#include <utility>
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#ifdef BOOST_MSVC
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# pragma warning(push)
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# pragma warning(disable: 4702) // unreachable code (return after domain_error throw).
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#endif
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namespace boost { namespace math {
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template <class RealType = double, class Policy = policies::policy<> >
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class students_t_distribution
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{
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public:
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typedef RealType value_type;
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typedef Policy policy_type;
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students_t_distribution(RealType df) : df_(df)
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{ // Constructor.
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RealType result;
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detail::check_df_gt0_to_inf( // Checks that df > 0 or df == inf.
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"boost::math::students_t_distribution<%1%>::students_t_distribution", df_, &result, Policy());
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} // students_t_distribution
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RealType degrees_of_freedom()const
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{
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return df_;
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}
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// Parameter estimation:
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static RealType find_degrees_of_freedom(
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RealType difference_from_mean,
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RealType alpha,
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RealType beta,
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RealType sd,
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RealType hint = 100);
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private:
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// Data member:
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RealType df_; // degrees of freedom is a real number > 0 or +infinity.
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};
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typedef students_t_distribution<double> students_t; // Convenience typedef for double version.
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template <class RealType, class Policy>
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inline const std::pair<RealType, RealType> range(const students_t_distribution<RealType, Policy>& /*dist*/)
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{ // Range of permissible values for random variable x.
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// Now including infinity.
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using boost::math::tools::max_value;
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//return std::pair<RealType, RealType>(-max_value<RealType>(), max_value<RealType>());
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return std::pair<RealType, RealType>(((::std::numeric_limits<RealType>::is_specialized & ::std::numeric_limits<RealType>::has_infinity) ? -std::numeric_limits<RealType>::infinity() : -max_value<RealType>()), ((::std::numeric_limits<RealType>::is_specialized & ::std::numeric_limits<RealType>::has_infinity) ? +std::numeric_limits<RealType>::infinity() : +max_value<RealType>()));
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}
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template <class RealType, class Policy>
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inline const std::pair<RealType, RealType> support(const students_t_distribution<RealType, Policy>& /*dist*/)
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{ // Range of supported values for random variable x.
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// Now including infinity.
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// This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.
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using boost::math::tools::max_value;
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//return std::pair<RealType, RealType>(-max_value<RealType>(), max_value<RealType>());
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return std::pair<RealType, RealType>(((::std::numeric_limits<RealType>::is_specialized & ::std::numeric_limits<RealType>::has_infinity) ? -std::numeric_limits<RealType>::infinity() : -max_value<RealType>()), ((::std::numeric_limits<RealType>::is_specialized & ::std::numeric_limits<RealType>::has_infinity) ? +std::numeric_limits<RealType>::infinity() : +max_value<RealType>()));
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}
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template <class RealType, class Policy>
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inline RealType pdf(const students_t_distribution<RealType, Policy>& dist, const RealType& x)
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{
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BOOST_FPU_EXCEPTION_GUARD
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BOOST_MATH_STD_USING // for ADL of std functions.
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RealType error_result;
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if(false == detail::check_x_not_NaN(
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"boost::math::pdf(const students_t_distribution<%1%>&, %1%)", x, &error_result, Policy()))
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return error_result;
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RealType df = dist.degrees_of_freedom();
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if(false == detail::check_df_gt0_to_inf( // Check that df > 0 or == +infinity.
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"boost::math::pdf(const students_t_distribution<%1%>&, %1%)", df, &error_result, Policy()))
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return error_result;
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RealType result;
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if ((boost::math::isinf)(x))
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{ // - or +infinity.
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result = static_cast<RealType>(0);
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return result;
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}
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RealType limit = policies::get_epsilon<RealType, Policy>();
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// Use policies so that if policy requests lower precision,
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// then get the normal distribution approximation earlier.
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limit = static_cast<RealType>(1) / limit; // 1/eps
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// for 64-bit double 1/eps = 4503599627370496
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if (df > limit)
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{ // Special case for really big degrees_of_freedom > 1 / eps
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// - use normal distribution which is much faster and more accurate.
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normal_distribution<RealType, Policy> n(0, 1);
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result = pdf(n, x);
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}
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else
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{ //
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RealType basem1 = x * x / df;
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if(basem1 < 0.125)
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{
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result = exp(-boost::math::log1p(basem1, Policy()) * (1+df) / 2);
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}
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else
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{
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result = pow(1 / (1 + basem1), (df + 1) / 2);
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}
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result /= sqrt(df) * boost::math::beta(df / 2, RealType(0.5f), Policy());
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}
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return result;
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} // pdf
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template <class RealType, class Policy>
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inline RealType cdf(const students_t_distribution<RealType, Policy>& dist, const RealType& x)
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{
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RealType error_result;
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// degrees_of_freedom > 0 or infinity check:
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RealType df = dist.degrees_of_freedom();
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if (false == detail::check_df_gt0_to_inf( // Check that df > 0 or == +infinity.
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"boost::math::cdf(const students_t_distribution<%1%>&, %1%)", df, &error_result, Policy()))
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{
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return error_result;
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}
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// Check for bad x first.
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if(false == detail::check_x_not_NaN(
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"boost::math::cdf(const students_t_distribution<%1%>&, %1%)", x, &error_result, Policy()))
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{
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return error_result;
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}
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if (x == 0)
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{ // Special case with exact result.
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return static_cast<RealType>(0.5);
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}
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if ((boost::math::isinf)(x))
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{ // x == - or + infinity, regardless of df.
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return ((x < 0) ? static_cast<RealType>(0) : static_cast<RealType>(1));
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}
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RealType limit = policies::get_epsilon<RealType, Policy>();
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// Use policies so that if policy requests lower precision,
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// then get the normal distribution approximation earlier.
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limit = static_cast<RealType>(1) / limit; // 1/eps
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// for 64-bit double 1/eps = 4503599627370496
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if (df > limit)
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{ // Special case for really big degrees_of_freedom > 1 / eps (perhaps infinite?)
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// - use normal distribution which is much faster and more accurate.
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normal_distribution<RealType, Policy> n(0, 1);
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RealType result = cdf(n, x);
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return result;
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}
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else
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{ // normal df case.
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//
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// Calculate probability of Student's t using the incomplete beta function.
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// probability = ibeta(degrees_of_freedom / 2, 1/2, degrees_of_freedom / (degrees_of_freedom + t*t))
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//
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// However when t is small compared to the degrees of freedom, that formula
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// suffers from rounding error, use the identity formula to work around
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// the problem:
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//
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// I[x](a,b) = 1 - I[1-x](b,a)
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//
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// and:
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//
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// x = df / (df + t^2)
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//
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// so:
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//
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// 1 - x = t^2 / (df + t^2)
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//
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RealType x2 = x * x;
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RealType probability;
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if(df > 2 * x2)
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{
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RealType z = x2 / (df + x2);
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probability = ibetac(static_cast<RealType>(0.5), df / 2, z, Policy()) / 2;
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}
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else
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{
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RealType z = df / (df + x2);
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probability = ibeta(df / 2, static_cast<RealType>(0.5), z, Policy()) / 2;
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}
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return (x > 0 ? 1 - probability : probability);
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}
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} // cdf
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template <class RealType, class Policy>
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inline RealType quantile(const students_t_distribution<RealType, Policy>& dist, const RealType& p)
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{
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BOOST_MATH_STD_USING // for ADL of std functions
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//
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// Obtain parameters:
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RealType probability = p;
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// Check for domain errors:
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RealType df = dist.degrees_of_freedom();
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static const char* function = "boost::math::quantile(const students_t_distribution<%1%>&, %1%)";
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RealType error_result;
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if(false == (detail::check_df_gt0_to_inf( // Check that df > 0 or == +infinity.
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function, df, &error_result, Policy())
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&& detail::check_probability(function, probability, &error_result, Policy())))
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return error_result;
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// Special cases, regardless of degrees_of_freedom.
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if (probability == 0)
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return -policies::raise_overflow_error<RealType>(function, 0, Policy());
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if (probability == 1)
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return policies::raise_overflow_error<RealType>(function, 0, Policy());
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if (probability == static_cast<RealType>(0.5))
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return 0; //
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//
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#if 0
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// This next block is disabled in favour of a faster method than
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// incomplete beta inverse, but code retained for future reference:
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//
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// Calculate quantile of Student's t using the incomplete beta function inverse:
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probability = (probability > 0.5) ? 1 - probability : probability;
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RealType t, x, y;
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x = ibeta_inv(degrees_of_freedom / 2, RealType(0.5), 2 * probability, &y);
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if(degrees_of_freedom * y > tools::max_value<RealType>() * x)
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t = tools::overflow_error<RealType>(function);
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else
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t = sqrt(degrees_of_freedom * y / x);
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//
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// Figure out sign based on the size of p:
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//
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if(p < 0.5)
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t = -t;
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return t;
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#endif
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//
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// Depending on how many digits RealType has, this may forward
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// to the incomplete beta inverse as above. Otherwise uses a
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// faster method that is accurate to ~15 digits everywhere
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// and a couple of epsilon at double precision and in the central
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// region where most use cases will occur...
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//
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return boost::math::detail::fast_students_t_quantile(df, probability, Policy());
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} // quantile
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template <class RealType, class Policy>
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inline RealType cdf(const complemented2_type<students_t_distribution<RealType, Policy>, RealType>& c)
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{
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return cdf(c.dist, -c.param);
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}
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template <class RealType, class Policy>
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inline RealType quantile(const complemented2_type<students_t_distribution<RealType, Policy>, RealType>& c)
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{
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return -quantile(c.dist, c.param);
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}
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//
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// Parameter estimation follows:
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//
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namespace detail{
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//
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// Functors for finding degrees of freedom:
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//
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template <class RealType, class Policy>
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struct sample_size_func
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{
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sample_size_func(RealType a, RealType b, RealType s, RealType d)
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: alpha(a), beta(b), ratio(s*s/(d*d)) {}
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RealType operator()(const RealType& df)
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{
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if(df <= tools::min_value<RealType>())
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{ //
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return 1;
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}
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students_t_distribution<RealType, Policy> t(df);
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RealType qa = quantile(complement(t, alpha));
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RealType qb = quantile(complement(t, beta));
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qa += qb;
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qa *= qa;
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qa *= ratio;
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qa -= (df + 1);
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return qa;
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}
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RealType alpha, beta, ratio;
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};
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} // namespace detail
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template <class RealType, class Policy>
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RealType students_t_distribution<RealType, Policy>::find_degrees_of_freedom(
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RealType difference_from_mean,
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RealType alpha,
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RealType beta,
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RealType sd,
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RealType hint)
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{
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static const char* function = "boost::math::students_t_distribution<%1%>::find_degrees_of_freedom";
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//
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// Check for domain errors:
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//
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RealType error_result;
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if(false == detail::check_probability(
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function, alpha, &error_result, Policy())
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&& detail::check_probability(function, beta, &error_result, Policy()))
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return error_result;
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if(hint <= 0)
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hint = 1;
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detail::sample_size_func<RealType, Policy> f(alpha, beta, sd, difference_from_mean);
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tools::eps_tolerance<RealType> tol(policies::digits<RealType, Policy>());
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boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();
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std::pair<RealType, RealType> r = tools::bracket_and_solve_root(f, hint, RealType(2), false, tol, max_iter, Policy());
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RealType result = r.first + (r.second - r.first) / 2;
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if(max_iter >= policies::get_max_root_iterations<Policy>())
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{
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return policies::raise_evaluation_error<RealType>(function, "Unable to locate solution in a reasonable time:"
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" either there is no answer to how many degrees of freedom are required"
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" or the answer is infinite. Current best guess is %1%", result, Policy());
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}
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return result;
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}
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template <class RealType, class Policy>
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inline RealType mode(const students_t_distribution<RealType, Policy>& /*dist*/)
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{
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// Assume no checks on degrees of freedom are useful (unlike mean).
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return 0; // Always zero by definition.
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}
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template <class RealType, class Policy>
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inline RealType median(const students_t_distribution<RealType, Policy>& /*dist*/)
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{
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// Assume no checks on degrees of freedom are useful (unlike mean).
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return 0; // Always zero by definition.
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}
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// See section 5.1 on moments at http://en.wikipedia.org/wiki/Student%27s_t-distribution
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template <class RealType, class Policy>
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inline RealType mean(const students_t_distribution<RealType, Policy>& dist)
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{ // Revised for https://svn.boost.org/trac/boost/ticket/7177
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RealType df = dist.degrees_of_freedom();
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if(((boost::math::isnan)(df)) || (df <= 1) )
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{ // mean is undefined for moment <= 1!
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return policies::raise_domain_error<RealType>(
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"boost::math::mean(students_t_distribution<%1%> const&, %1%)",
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"Mean is undefined for degrees of freedom < 1 but got %1%.", df, Policy());
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return std::numeric_limits<RealType>::quiet_NaN();
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}
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return 0;
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} // mean
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template <class RealType, class Policy>
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inline RealType variance(const students_t_distribution<RealType, Policy>& dist)
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{ // http://en.wikipedia.org/wiki/Student%27s_t-distribution
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// Revised for https://svn.boost.org/trac/boost/ticket/7177
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RealType df = dist.degrees_of_freedom();
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if ((boost::math::isnan)(df) || (df <= 2))
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{ // NaN or undefined for <= 2.
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return policies::raise_domain_error<RealType>(
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"boost::math::variance(students_t_distribution<%1%> const&, %1%)",
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"variance is undefined for degrees of freedom <= 2, but got %1%.",
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df, Policy());
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return std::numeric_limits<RealType>::quiet_NaN(); // Undefined.
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}
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if ((boost::math::isinf)(df))
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{ // +infinity.
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return 1;
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}
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RealType limit = policies::get_epsilon<RealType, Policy>();
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// Use policies so that if policy requests lower precision,
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// then get the normal distribution approximation earlier.
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limit = static_cast<RealType>(1) / limit; // 1/eps
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// for 64-bit double 1/eps = 4503599627370496
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if (df > limit)
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{ // Special case for really big degrees_of_freedom > 1 / eps.
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return 1;
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}
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else
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{
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return df / (df - 2);
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}
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} // variance
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template <class RealType, class Policy>
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inline RealType skewness(const students_t_distribution<RealType, Policy>& dist)
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{
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RealType df = dist.degrees_of_freedom();
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if( ((boost::math::isnan)(df)) || (dist.degrees_of_freedom() <= 3))
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{ // Undefined for moment k = 3.
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return policies::raise_domain_error<RealType>(
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"boost::math::skewness(students_t_distribution<%1%> const&, %1%)",
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"Skewness is undefined for degrees of freedom <= 3, but got %1%.",
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dist.degrees_of_freedom(), Policy());
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return std::numeric_limits<RealType>::quiet_NaN();
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}
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return 0; // For all valid df, including infinity.
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} // skewness
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template <class RealType, class Policy>
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inline RealType kurtosis(const students_t_distribution<RealType, Policy>& dist)
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{
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RealType df = dist.degrees_of_freedom();
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if(((boost::math::isnan)(df)) || (df <= 4))
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{ // Undefined or infinity for moment k = 4.
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return policies::raise_domain_error<RealType>(
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"boost::math::kurtosis(students_t_distribution<%1%> const&, %1%)",
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"Kurtosis is undefined for degrees of freedom <= 4, but got %1%.",
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df, Policy());
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return std::numeric_limits<RealType>::quiet_NaN(); // Undefined.
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}
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if ((boost::math::isinf)(df))
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{ // +infinity.
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return 3;
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}
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RealType limit = policies::get_epsilon<RealType, Policy>();
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// Use policies so that if policy requests lower precision,
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// then get the normal distribution approximation earlier.
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limit = static_cast<RealType>(1) / limit; // 1/eps
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// for 64-bit double 1/eps = 4503599627370496
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if (df > limit)
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{ // Special case for really big degrees_of_freedom > 1 / eps.
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return 3;
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}
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else
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{
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//return 3 * (df - 2) / (df - 4); re-arranged to
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return 6 / (df - 4) + 3;
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}
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} // kurtosis
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template <class RealType, class Policy>
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inline RealType kurtosis_excess(const students_t_distribution<RealType, Policy>& dist)
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{
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// see http://mathworld.wolfram.com/Kurtosis.html
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RealType df = dist.degrees_of_freedom();
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if(((boost::math::isnan)(df)) || (df <= 4))
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{ // Undefined or infinity for moment k = 4.
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return policies::raise_domain_error<RealType>(
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"boost::math::kurtosis_excess(students_t_distribution<%1%> const&, %1%)",
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"Kurtosis_excess is undefined for degrees of freedom <= 4, but got %1%.",
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df, Policy());
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return std::numeric_limits<RealType>::quiet_NaN(); // Undefined.
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}
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if ((boost::math::isinf)(df))
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{ // +infinity.
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return 0;
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}
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RealType limit = policies::get_epsilon<RealType, Policy>();
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// Use policies so that if policy requests lower precision,
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// then get the normal distribution approximation earlier.
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limit = static_cast<RealType>(1) / limit; // 1/eps
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// for 64-bit double 1/eps = 4503599627370496
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if (df > limit)
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{ // Special case for really big degrees_of_freedom > 1 / eps.
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return 0;
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}
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else
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{
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return 6 / (df - 4);
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}
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}
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template <class RealType, class Policy>
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inline RealType entropy(const students_t_distribution<RealType, Policy>& dist)
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{
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using std::log;
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using std::sqrt;
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RealType v = dist.degrees_of_freedom();
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RealType vp1 = (v+1)/2;
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RealType vd2 = v/2;
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return vp1*(digamma(vp1) - digamma(vd2)) + log(sqrt(v)*beta(vd2, RealType(1)/RealType(2)));
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}
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} // namespace math
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} // namespace boost
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#ifdef BOOST_MSVC
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# pragma warning(pop)
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#endif
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// This include must be at the end, *after* the accessors
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// for this distribution have been defined, in order to
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// keep compilers that support two-phase lookup happy.
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#include <boost/math/distributions/detail/derived_accessors.hpp>
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#endif // BOOST_STATS_STUDENTS_T_HPP
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