Chris@16: // Copyright John Maddock 2010.
Chris@16: // Copyright Paul A. Bristow 2010.
Chris@16: 
Chris@16: // Use, modification and distribution are subject to the
Chris@16: // Boost Software License, Version 1.0.
Chris@16: // (See accompanying file LICENSE_1_0.txt
Chris@16: // or copy at http://www.boost.org/LICENSE_1_0.txt)
Chris@16: 
Chris@16: #ifndef BOOST_MATH_DISTRIBUTIONS_INVERSE_CHI_SQUARED_HPP
Chris@16: #define BOOST_MATH_DISTRIBUTIONS_INVERSE_CHI_SQUARED_HPP
Chris@16: 
Chris@16: #include <boost/math/distributions/fwd.hpp>
Chris@16: #include <boost/math/special_functions/gamma.hpp> // for incomplete beta.
Chris@16: #include <boost/math/distributions/complement.hpp> // for complements.
Chris@16: #include <boost/math/distributions/detail/common_error_handling.hpp> // for error checks.
Chris@16: #include <boost/math/special_functions/fpclassify.hpp> // for isfinite
Chris@16: 
Chris@16: // See http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution
Chris@16: // for definitions of this scaled version.
Chris@16: // See http://en.wikipedia.org/wiki/Inverse-chi-square_distribution
Chris@16: // for unscaled version.
Chris@16: 
Chris@16: // http://reference.wolfram.com/mathematica/ref/InverseChiSquareDistribution.html
Chris@16: // Weisstein, Eric W. "Inverse Chi-Squared Distribution." From MathWorld--A Wolfram Web Resource.
Chris@16: // http://mathworld.wolfram.com/InverseChi-SquaredDistribution.html
Chris@16: 
Chris@16: #include <utility>
Chris@16: 
Chris@16: namespace boost{ namespace math{
Chris@16: 
Chris@16: namespace detail
Chris@16: {
Chris@16:   template <class RealType, class Policy>
Chris@16:   inline bool check_inverse_chi_squared( // Check both distribution parameters.
Chris@16:         const char* function,
Chris@16:         RealType degrees_of_freedom, // degrees_of_freedom (aka nu).
Chris@16:         RealType scale,  // scale (aka sigma^2)
Chris@16:         RealType* result,
Chris@16:         const Policy& pol)
Chris@16:   {
Chris@16:      return check_scale(function, scale, result, pol)
Chris@16:        && check_df(function, degrees_of_freedom,
Chris@16:        result, pol);
Chris@16:   } // bool check_inverse_chi_squared
Chris@16: } // namespace detail
Chris@16: 
Chris@16: template <class RealType = double, class Policy = policies::policy<> >
Chris@16: class inverse_chi_squared_distribution
Chris@16: {
Chris@16: public:
Chris@16:    typedef RealType value_type;
Chris@16:    typedef Policy policy_type;
Chris@16: 
Chris@16:    inverse_chi_squared_distribution(RealType df, RealType l_scale) : m_df(df), m_scale (l_scale)
Chris@16:    {
Chris@16:       RealType result;
Chris@16:       detail::check_df(
Chris@16:          "boost::math::inverse_chi_squared_distribution<%1%>::inverse_chi_squared_distribution",
Chris@16:          m_df, &result, Policy())
Chris@16:          && detail::check_scale(
Chris@16: "boost::math::inverse_chi_squared_distribution<%1%>::inverse_chi_squared_distribution",
Chris@16:          m_scale, &result,  Policy());
Chris@16:    } // inverse_chi_squared_distribution constructor 
Chris@16: 
Chris@16:    inverse_chi_squared_distribution(RealType df = 1) : m_df(df)
Chris@16:    {
Chris@16:       RealType result;
Chris@16:       m_scale = 1 / m_df ; // Default scale = 1 / degrees of freedom (Wikipedia definition 1).
Chris@16:       detail::check_df(
Chris@16:          "boost::math::inverse_chi_squared_distribution<%1%>::inverse_chi_squared_distribution",
Chris@16:          m_df, &result, Policy());
Chris@16:    } // inverse_chi_squared_distribution
Chris@16: 
Chris@16:    RealType degrees_of_freedom()const
Chris@16:    {
Chris@16:       return m_df; // aka nu
Chris@16:    }
Chris@16:    RealType scale()const
Chris@16:    {
Chris@16:       return m_scale;  // aka xi
Chris@16:    }
Chris@16: 
Chris@16:    // Parameter estimation:  NOT implemented yet.
Chris@16:    //static RealType find_degrees_of_freedom(
Chris@16:    //   RealType difference_from_variance,
Chris@16:    //   RealType alpha,
Chris@16:    //   RealType beta,
Chris@16:    //   RealType variance,
Chris@16:    //   RealType hint = 100);
Chris@16: 
Chris@16: private:
Chris@16:    // Data members:
Chris@16:    RealType m_df;  // degrees of freedom are treated as a real number.
Chris@16:    RealType m_scale;  // distribution scale.
Chris@16: 
Chris@16: }; // class chi_squared_distribution
Chris@16: 
Chris@16: typedef inverse_chi_squared_distribution<double> inverse_chi_squared;
Chris@16: 
Chris@16: template <class RealType, class Policy>
Chris@16: inline const std::pair<RealType, RealType> range(const inverse_chi_squared_distribution<RealType, Policy>& /*dist*/)
Chris@16: {  // Range of permissible values for random variable x.
Chris@16:    using boost::math::tools::max_value;
Chris@16:    return std::pair<RealType, RealType>(static_cast<RealType>(0), max_value<RealType>()); // 0 to + infinity.
Chris@16: }
Chris@16: 
Chris@16: template <class RealType, class Policy>
Chris@16: inline const std::pair<RealType, RealType> support(const inverse_chi_squared_distribution<RealType, Policy>& /*dist*/)
Chris@16: {  // Range of supported values for random variable x.
Chris@16:    // This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.
Chris@16:    return std::pair<RealType, RealType>(static_cast<RealType>(0), tools::max_value<RealType>()); // 0 to + infinity.
Chris@16: }
Chris@16: 
Chris@16: template <class RealType, class Policy>
Chris@16: RealType pdf(const inverse_chi_squared_distribution<RealType, Policy>& dist, const RealType& x)
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING  // for ADL of std functions.
Chris@16:    RealType df = dist.degrees_of_freedom();
Chris@16:    RealType scale = dist.scale();
Chris@16:    RealType error_result;
Chris@16: 
Chris@16:    static const char* function = "boost::math::pdf(const inverse_chi_squared_distribution<%1%>&, %1%)";
Chris@16: 
Chris@16:    if(false == detail::check_inverse_chi_squared
Chris@16:      (function, df, scale, &error_result, Policy())
Chris@16:      )
Chris@16:    { // Bad distribution.
Chris@16:       return error_result;
Chris@16:    }
Chris@16:    if((x < 0) || !(boost::math::isfinite)(x))
Chris@16:    { // Bad x.
Chris@16:       return policies::raise_domain_error<RealType>(
Chris@16:          function, "inverse Chi Square parameter was %1%, but must be >= 0 !", x, Policy());
Chris@16:    }
Chris@16: 
Chris@16:    if(x == 0)
Chris@16:    { // Treat as special case.
Chris@16:      return 0;
Chris@16:    }
Chris@16:    // Wikipedia scaled inverse chi sq (df, scale) related to inv gamma (df/2, df * scale /2) 
Chris@16:    // so use inverse gamma pdf with shape = df/2, scale df * scale /2 
Chris@16:    // RealType shape = df /2; // inv_gamma shape
Chris@16:    // RealType scale = df * scale/2; // inv_gamma scale
Chris@16:    // RealType result = gamma_p_derivative(shape, scale / x, Policy()) * scale / (x * x);
Chris@16:    RealType result = df * scale/2 / x;
Chris@16:    if(result < tools::min_value<RealType>())
Chris@16:       return 0; // Random variable is near enough infinite.
Chris@16:    result = gamma_p_derivative(df/2, result, Policy()) * df * scale/2;
Chris@16:    if(result != 0) // prevent 0 / 0,  gamma_p_derivative -> 0 faster than x^2
Chris@16:       result /= (x * x);
Chris@16:    return result;
Chris@16: } // pdf
Chris@16: 
Chris@16: template <class RealType, class Policy>
Chris@16: inline RealType cdf(const inverse_chi_squared_distribution<RealType, Policy>& dist, const RealType& x)
Chris@16: {
Chris@16:    static const char* function = "boost::math::cdf(const inverse_chi_squared_distribution<%1%>&, %1%)";
Chris@16:    RealType df = dist.degrees_of_freedom();
Chris@16:    RealType scale = dist.scale();
Chris@16:    RealType error_result;
Chris@16: 
Chris@16:    if(false ==
Chris@16:        detail::check_inverse_chi_squared(function, df, scale, &error_result, Policy())
Chris@16:      )
Chris@16:    { // Bad distribution.
Chris@16:       return error_result;
Chris@16:    }
Chris@16:    if((x < 0) || !(boost::math::isfinite)(x))
Chris@16:    { // Bad x.
Chris@16:       return policies::raise_domain_error<RealType>(
Chris@16:          function, "inverse Chi Square parameter was %1%, but must be >= 0 !", x, Policy());
Chris@16:    }
Chris@16:    if (x == 0)
Chris@16:    { // Treat zero as a special case.
Chris@16:      return 0;
Chris@16:    }
Chris@16:    // RealType shape = df /2; // inv_gamma shape,
Chris@16:    // RealType scale = df * scale/2; // inv_gamma scale,
Chris@16:    // result = boost::math::gamma_q(shape, scale / x, Policy()); // inverse_gamma code.
Chris@16:    return boost::math::gamma_q(df / 2, (df * (scale / 2)) / x, Policy());
Chris@16: } // cdf
Chris@16: 
Chris@16: template <class RealType, class Policy>
Chris@16: inline RealType quantile(const inverse_chi_squared_distribution<RealType, Policy>& dist, const RealType& p)
Chris@16: {
Chris@16:    using boost::math::gamma_q_inv;
Chris@16:    RealType df = dist.degrees_of_freedom();
Chris@16:    RealType scale = dist.scale();
Chris@16: 
Chris@16:    static const char* function = "boost::math::quantile(const inverse_chi_squared_distribution<%1%>&, %1%)";
Chris@16:    // Error check:
Chris@16:    RealType error_result;
Chris@16:    if(false == detail::check_df(
Chris@16:          function, df, &error_result, Policy())
Chris@16:          && detail::check_probability(
Chris@16:             function, p, &error_result, Policy()))
Chris@16:    {
Chris@16:       return error_result;
Chris@16:    }
Chris@16:    if(false == detail::check_probability(
Chris@16:             function, p, &error_result, Policy()))
Chris@16:    {
Chris@16:       return error_result;
Chris@16:    }
Chris@16:    // RealType shape = df /2; // inv_gamma shape,
Chris@16:    // RealType scale = df * scale/2; // inv_gamma scale,
Chris@16:    // result = scale / gamma_q_inv(shape, p, Policy());
Chris@16:       RealType result = gamma_q_inv(df /2, p, Policy());
Chris@16:       if(result == 0)
Chris@16:          return policies::raise_overflow_error<RealType, Policy>(function, "Random variable is infinite.", Policy());
Chris@16:       result = df * (scale / 2) / result;
Chris@16:       return result;
Chris@16: } // quantile
Chris@16: 
Chris@16: template <class RealType, class Policy>
Chris@16: inline RealType cdf(const complemented2_type<inverse_chi_squared_distribution<RealType, Policy>, RealType>& c)
Chris@16: {
Chris@16:    using boost::math::gamma_q_inv;
Chris@16:    RealType const& df = c.dist.degrees_of_freedom();
Chris@16:    RealType const& scale = c.dist.scale();
Chris@16:    RealType const& x = c.param;
Chris@16:    static const char* function = "boost::math::cdf(const inverse_chi_squared_distribution<%1%>&, %1%)";
Chris@16:    // Error check:
Chris@16:    RealType error_result;
Chris@16:    if(false == detail::check_df(
Chris@16:          function, df, &error_result, Policy()))
Chris@16:    {
Chris@16:       return error_result;
Chris@16:    }
Chris@16:    if (x == 0)
Chris@16:    { // Treat zero as a special case.
Chris@16:      return 1;
Chris@16:    }
Chris@16:    if((x < 0) || !(boost::math::isfinite)(x))
Chris@16:    {
Chris@16:       return policies::raise_domain_error<RealType>(
Chris@16:          function, "inverse Chi Square parameter was %1%, but must be > 0 !", x, Policy());
Chris@16:    }
Chris@16:    // RealType shape = df /2; // inv_gamma shape,
Chris@16:    // RealType scale = df * scale/2; // inv_gamma scale,
Chris@16:    // result = gamma_p(shape, scale/c.param, Policy()); use inv_gamma.
Chris@16: 
Chris@16:    return gamma_p(df / 2, (df * scale/2) / x, Policy()); // OK
Chris@16: } // cdf(complemented
Chris@16: 
Chris@16: template <class RealType, class Policy>
Chris@16: inline RealType quantile(const complemented2_type<inverse_chi_squared_distribution<RealType, Policy>, RealType>& c)
Chris@16: {
Chris@16:    using boost::math::gamma_q_inv;
Chris@16: 
Chris@16:    RealType const& df = c.dist.degrees_of_freedom();
Chris@16:    RealType const& scale = c.dist.scale();
Chris@16:    RealType const& q = c.param;
Chris@16:    static const char* function = "boost::math::quantile(const inverse_chi_squared_distribution<%1%>&, %1%)";
Chris@16:    // Error check:
Chris@16:    RealType error_result;
Chris@16:    if(false == detail::check_df(function, df, &error_result, Policy()))
Chris@16:    {
Chris@16:       return error_result;
Chris@16:    }
Chris@16:    if(false == detail::check_probability(function, q, &error_result, Policy()))
Chris@16:    {
Chris@16:       return error_result;
Chris@16:    }
Chris@16:    // RealType shape = df /2; // inv_gamma shape,
Chris@16:    // RealType scale = df * scale/2; // inv_gamma scale,
Chris@16:    // result = scale / gamma_p_inv(shape, q, Policy());  // using inv_gamma.
Chris@16:    RealType result = gamma_p_inv(df/2, q, Policy());
Chris@16:    if(result == 0)
Chris@16:       return policies::raise_overflow_error<RealType, Policy>(function, "Random variable is infinite.", Policy());
Chris@16:    result = (df * scale / 2) / result;
Chris@16:    return result;
Chris@16: } // quantile(const complement
Chris@16: 
Chris@16: template <class RealType, class Policy>
Chris@16: inline RealType mean(const inverse_chi_squared_distribution<RealType, Policy>& dist)
Chris@16: { // Mean of inverse Chi-Squared distribution.
Chris@16:    RealType df = dist.degrees_of_freedom();
Chris@16:    RealType scale = dist.scale();
Chris@16: 
Chris@16:    static const char* function = "boost::math::mean(const inverse_chi_squared_distribution<%1%>&)";
Chris@16:    if(df <= 2)
Chris@16:       return policies::raise_domain_error<RealType>(
Chris@16:          function,
Chris@16:          "inverse Chi-Squared distribution only has a mode for degrees of freedom > 2, but got degrees of freedom = %1%.",
Chris@16:          df, Policy());
Chris@16:   return (df * scale) / (df - 2);
Chris@16: } // mean
Chris@16: 
Chris@16: template <class RealType, class Policy>
Chris@16: inline RealType variance(const inverse_chi_squared_distribution<RealType, Policy>& dist)
Chris@16: { // Variance of inverse Chi-Squared distribution.
Chris@16:    RealType df = dist.degrees_of_freedom();
Chris@16:    RealType scale = dist.scale();
Chris@16:    static const char* function = "boost::math::variance(const inverse_chi_squared_distribution<%1%>&)";
Chris@16:    if(df <= 4)
Chris@16:    {
Chris@16:       return policies::raise_domain_error<RealType>(
Chris@16:          function,
Chris@16:          "inverse Chi-Squared distribution only has a variance for degrees of freedom > 4, but got degrees of freedom = %1%.",
Chris@16:          df, Policy());
Chris@16:    }
Chris@16:    return 2 * df * df * scale * scale / ((df - 2)*(df - 2) * (df - 4));
Chris@16: } // variance
Chris@16: 
Chris@16: template <class RealType, class Policy>
Chris@16: inline RealType mode(const inverse_chi_squared_distribution<RealType, Policy>& dist)
Chris@16: { // mode is not defined in Mathematica.
Chris@16:   // See Discussion section http://en.wikipedia.org/wiki/Talk:Scaled-inverse-chi-square_distribution
Chris@16:   // for origin of the formula used below.
Chris@16: 
Chris@16:    RealType df = dist.degrees_of_freedom();
Chris@16:    RealType scale = dist.scale();
Chris@16:    static const char* function = "boost::math::mode(const inverse_chi_squared_distribution<%1%>&)";
Chris@16:    if(df < 0)
Chris@16:       return policies::raise_domain_error<RealType>(
Chris@16:          function,
Chris@16:          "inverse Chi-Squared distribution only has a mode for degrees of freedom >= 0, but got degrees of freedom = %1%.",
Chris@16:          df, Policy());
Chris@16:    return (df * scale) / (df + 2);
Chris@16: }
Chris@16: 
Chris@16: //template <class RealType, class Policy>
Chris@16: //inline RealType median(const inverse_chi_squared_distribution<RealType, Policy>& dist)
Chris@16: //{ // Median is given by Quantile[dist, 1/2]
Chris@16: //   RealType df = dist.degrees_of_freedom();
Chris@16: //   if(df <= 1)
Chris@16: //      return tools::domain_error<RealType>(
Chris@16: //         BOOST_CURRENT_FUNCTION,
Chris@16: //         "The inverse_Chi-Squared distribution only has a median for degrees of freedom >= 0, but got degrees of freedom = %1%.",
Chris@16: //         df);
Chris@16: //   return df;
Chris@16: //}
Chris@16: // Now implemented via quantile(half) in derived accessors.
Chris@16: 
Chris@16: template <class RealType, class Policy>
Chris@16: inline RealType skewness(const inverse_chi_squared_distribution<RealType, Policy>& dist)
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING // For ADL
Chris@16:    RealType df = dist.degrees_of_freedom();
Chris@16:    static const char* function = "boost::math::skewness(const inverse_chi_squared_distribution<%1%>&)";
Chris@16:    if(df <= 6)
Chris@16:       return policies::raise_domain_error<RealType>(
Chris@16:          function,
Chris@16:          "inverse Chi-Squared distribution only has a skewness for degrees of freedom > 6, but got degrees of freedom = %1%.",
Chris@16:          df, Policy());
Chris@16: 
Chris@16:    return 4 * sqrt (2 * (df - 4)) / (df - 6);  // Not a function of scale.
Chris@16: }
Chris@16: 
Chris@16: template <class RealType, class Policy>
Chris@16: inline RealType kurtosis(const inverse_chi_squared_distribution<RealType, Policy>& dist)
Chris@16: {
Chris@16:    RealType df = dist.degrees_of_freedom();
Chris@16:    static const char* function = "boost::math::kurtosis(const inverse_chi_squared_distribution<%1%>&)";
Chris@16:    if(df <= 8)
Chris@16:       return policies::raise_domain_error<RealType>(
Chris@16:          function,
Chris@16:          "inverse Chi-Squared distribution only has a kurtosis for degrees of freedom > 8, but got degrees of freedom = %1%.",
Chris@16:          df, Policy());
Chris@16: 
Chris@16:    return kurtosis_excess(dist) + 3;
Chris@16: }
Chris@16: 
Chris@16: template <class RealType, class Policy>
Chris@16: inline RealType kurtosis_excess(const inverse_chi_squared_distribution<RealType, Policy>& dist)
Chris@16: {
Chris@16:    RealType df = dist.degrees_of_freedom();
Chris@16:    static const char* function = "boost::math::kurtosis(const inverse_chi_squared_distribution<%1%>&)";
Chris@16:    if(df <= 8)
Chris@16:       return policies::raise_domain_error<RealType>(
Chris@16:          function,
Chris@16:          "inverse Chi-Squared distribution only has a kurtosis excess for degrees of freedom > 8, but got degrees of freedom = %1%.",
Chris@16:          df, Policy());
Chris@16: 
Chris@16:    return 12 * (5 * df - 22) / ((df - 6 )*(df - 8));  // Not a function of scale.
Chris@16: }
Chris@16: 
Chris@16: //
Chris@16: // Parameter estimation comes last:
Chris@16: //
Chris@16: 
Chris@16: } // namespace math
Chris@16: } // namespace boost
Chris@16: 
Chris@16: // This include must be at the end, *after* the accessors
Chris@16: // for this distribution have been defined, in order to
Chris@16: // keep compilers that support two-phase lookup happy.
Chris@16: #include <boost/math/distributions/detail/derived_accessors.hpp>
Chris@16: 
Chris@16: #endif // BOOST_MATH_DISTRIBUTIONS_INVERSE_CHI_SQUARED_HPP