text: // Copyright John Maddock 2006, 2007.
  // Copyright Paul A. Bristow 2008, 2010.
  
  // Use, modification and distribution are subject to the
  // Boost Software License, Version 1.0.
  // (See accompanying file LICENSE_1_0.txt
  // or copy at http://www.boost.org/LICENSE_1_0.txt)
  
  #ifndef BOOST_MATH_DISTRIBUTIONS_CHI_SQUARED_HPP
  #define BOOST_MATH_DISTRIBUTIONS_CHI_SQUARED_HPP
  
  #include <boost/math/distributions/fwd.hpp>
  #include <boost/math/special_functions/gamma.hpp> // for incomplete beta.
  #include <boost/math/distributions/complement.hpp> // complements
  #include <boost/math/distributions/detail/common_error_handling.hpp> // error checks
  #include <boost/math/special_functions/fpclassify.hpp>
  
  #include <utility>
  
  namespace boost{ namespace math{
  
  template <class RealType = double, class Policy = policies::policy<> >
  class chi_squared_distribution
  {
  public:
     typedef RealType value_type;
     typedef Policy policy_type;
  
     chi_squared_distribution(RealType i) : m_df(i)
     {
        RealType result;
        detail::check_df(
           "boost::math::chi_squared_distribution<%1%>::chi_squared_distribution", m_df, &result, Policy());
     } // chi_squared_distribution
  
     RealType degrees_of_freedom()const
     {
        return m_df;
     }
  
     // Parameter estimation:
     static RealType find_degrees_of_freedom(
        RealType difference_from_variance,
        RealType alpha,
        RealType beta,
        RealType variance,
        RealType hint = 100);
  
  private:
     //
     // Data member:
     //
     RealType m_df; // degrees of freedom is a positive real number.
  }; // class chi_squared_distribution
  
  typedef chi_squared_distribution<double> chi_squared;
  
  #ifdef BOOST_MSVC
  #pragma warning(push)
  #pragma warning(disable:4127)
  #endif
  
  template <class RealType, class Policy>
  inline const std::pair<RealType, RealType> range(const chi_squared_distribution<RealType, Policy>& /*dist*/)
  { // Range of permissible values for random variable x.
    if (std::numeric_limits<RealType>::has_infinity)
    { 
      return std::pair<RealType, RealType>(static_cast<RealType>(0), std::numeric_limits<RealType>::infinity()); // 0 to + infinity.
    }
    else
    {
      using boost::math::tools::max_value;
      return std::pair<RealType, RealType>(static_cast<RealType>(0), max_value<RealType>()); // 0 to + max.
    }
  }
  
  #ifdef BOOST_MSVC
  #pragma warning(pop)
  #endif
  
  template <class RealType, class Policy>
  inline const std::pair<RealType, RealType> support(const chi_squared_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.
     return std::pair<RealType, RealType>(static_cast<RealType>(0), tools::max_value<RealType>()); // 0 to + infinity.
  }
  
  template <class RealType, class Policy>
  RealType pdf(const chi_squared_distribution<RealType, Policy>& dist, const RealType& chi_square)
  {
     BOOST_MATH_STD_USING  // for ADL of std functions
     RealType degrees_of_freedom = dist.degrees_of_freedom();
     // Error check:
     RealType error_result;
  
     static const char* function = "boost::math::pdf(const chi_squared_distribution<%1%>&, %1%)";
  
     if(false == detail::check_df(
           function, degrees_of_freedom, &error_result, Policy()))
        return error_result;
  
     if((chi_square < 0) || !(boost::math::isfinite)(chi_square))
     {
        return policies::raise_domain_error<RealType>(
           function, "Chi Square parameter was %1%, but must be > 0 !", chi_square, Policy());
     }
  
     if(chi_square == 0)
     {
        // Handle special cases:
        if(degrees_of_freedom < 2)
        {
           return policies::raise_overflow_error<RealType>(
              function, 0, Policy());
        }
        else if(degrees_of_freedom == 2)
        {
           return 0.5f;
        }
        else
        {
           return 0;
        }
     }
  
     return gamma_p_derivative(degrees_of_freedom / 2, chi_square / 2, Policy()) / 2;
  } // pdf
  
  template <class RealType, class Policy>
  inline RealType cdf(const chi_squared_distribution<RealType, Policy>& dist, const RealType& chi_square)
  {
     RealType degrees_of_freedom = dist.degrees_of_freedom();
     // Error check:
     RealType error_result;
     static const char* function = "boost::math::cdf(const chi_squared_distribution<%1%>&, %1%)";
  
     if(false == detail::check_df(
           function, degrees_of_freedom, &error_result, Policy()))
        return error_result;
  
     if((chi_square < 0) || !(boost::math::isfinite)(chi_square))
     {
        return policies::raise_domain_error<RealType>(
           function, "Chi Square parameter was %1%, but must be > 0 !", chi_square, Policy());
     }
  
     return boost::math::gamma_p(degrees_of_freedom / 2, chi_square / 2, Policy());
  } // cdf
  
  template <class RealType, class Policy>
  inline RealType quantile(const chi_squared_distribution<RealType, Policy>& dist, const RealType& p)
  {
     RealType degrees_of_freedom = dist.degrees_of_freedom();
     static const char* function = "boost::math::quantile(const chi_squared_distribution<%1%>&, %1%)";
     // Error check:
     RealType error_result;
     if(false ==
       (
         detail::check_df(function, degrees_of_freedom, &error_result, Policy())
         && detail::check_probability(function, p, &error_result, Policy()))
       )
       return error_result;
  
     return 2 * boost::math::gamma_p_inv(degrees_of_freedom / 2, p, Policy());
  } // quantile
  
  template <class RealType, class Policy>
  inline RealType cdf(const complemented2_type<chi_squared_distribution<RealType, Policy>, RealType>& c)
  {
     RealType const& degrees_of_freedom = c.dist.degrees_of_freedom();
     RealType const& chi_square = c.param;
     static const char* function = "boost::math::cdf(const chi_squared_distribution<%1%>&, %1%)";
     // Error check:
     RealType error_result;
     if(false == detail::check_df(
           function, degrees_of_freedom, &error_result, Policy()))
        return error_result;
  
     if((chi_square < 0) || !(boost::math::isfinite)(chi_square))
     {
        return policies::raise_domain_error<RealType>(
           function, "Chi Square parameter was %1%, but must be > 0 !", chi_square, Policy());
     }
  
     return boost::math::gamma_q(degrees_of_freedom / 2, chi_square / 2, Policy());
  }
  
  template <class RealType, class Policy>
  inline RealType quantile(const complemented2_type<chi_squared_distribution<RealType, Policy>, RealType>& c)
  {
     RealType const& degrees_of_freedom = c.dist.degrees_of_freedom();
     RealType const& q = c.param;
     static const char* function = "boost::math::quantile(const chi_squared_distribution<%1%>&, %1%)";
     // Error check:
     RealType error_result;
     if(false == (
       detail::check_df(function, degrees_of_freedom, &error_result, Policy())
       && detail::check_probability(function, q, &error_result, Policy()))
       )
      return error_result;
  
     return 2 * boost::math::gamma_q_inv(degrees_of_freedom / 2, q, Policy());
  }
  
  template <class RealType, class Policy>
  inline RealType mean(const chi_squared_distribution<RealType, Policy>& dist)
  { // Mean of Chi-Squared distribution = v.
    return dist.degrees_of_freedom();
  } // mean
  
  template <class RealType, class Policy>
  inline RealType variance(const chi_squared_distribution<RealType, Policy>& dist)
  { // Variance of Chi-Squared distribution = 2v.
    return 2 * dist.degrees_of_freedom();
  } // variance
  
  template <class RealType, class Policy>
  inline RealType mode(const chi_squared_distribution<RealType, Policy>& dist)
  {
     RealType df = dist.degrees_of_freedom();
     static const char* function = "boost::math::mode(const chi_squared_distribution<%1%>&)";
     // Most sources only define mode for df >= 2,
     // but for 0 <= df <= 2, the pdf maximum actually occurs at random variate = 0;
     // So one could extend the definition of mode thus:
     //if(df < 0)
     //{
     //   return policies::raise_domain_error<RealType>(
     //      function,
     //      "Chi-Squared distribution only has a mode for degrees of freedom >= 0, but got degrees of freedom = %1%.",
     //      df, Policy());
     //}
     //return (df <= 2) ? 0 : df - 2;
  
     if(df < 2)
        return policies::raise_domain_error<RealType>(
           function,
           "Chi-Squared distribution only has a mode for degrees of freedom >= 2, but got degrees of freedom = %1%.",
           df, Policy());
     return df - 2;
  }
  
  //template <class RealType, class Policy>
  //inline RealType median(const chi_squared_distribution<RealType, Policy>& dist)
  //{ // Median is given by Quantile[dist, 1/2]
  //   RealType df = dist.degrees_of_freedom();
  //   if(df <= 1)
  //      return tools::domain_error<RealType>(
  //         BOOST_CURRENT_FUNCTION,
  //         "The Chi-Squared distribution only has a mode for degrees of freedom >= 2, but got degrees of freedom = %1%.",
  //         df);
  //   return df - RealType(2)/3;
  //}
  // Now implemented via quantile(half) in derived accessors.
  
  template <class RealType, class Policy>
  inline RealType skewness(const chi_squared_distribution<RealType, Policy>& dist)
  {
     BOOST_MATH_STD_USING // For ADL
     RealType df = dist.degrees_of_freedom();
     return sqrt (8 / df);  // == 2 * sqrt(2 / df);
  }
  
  template <class RealType, class Policy>
  inline RealType kurtosis(const chi_squared_distribution<RealType, Policy>& dist)
  {
     RealType df = dist.degrees_of_freedom();
     return 3 + 12 / df;
  }
  
  template <class RealType, class Policy>
  inline RealType kurtosis_excess(const chi_squared_distribution<RealType, Policy>& dist)
  {
     RealType df = dist.degrees_of_freedom();
     return 12 / df;
  }
  
  //
  // Parameter estimation comes last:
  //
  namespace detail
  {
  
  template <class RealType, class Policy>
  struct df_estimator
  {
     df_estimator(RealType a, RealType b, RealType variance, RealType delta)
        : alpha(a), beta(b), ratio(delta/variance)
     { // Constructor
     }
  
     RealType operator()(const RealType& df)
     {
        if(df <= tools::min_value<RealType>())
           return 1;
        chi_squared_distribution<RealType, Policy> cs(df);
  
        RealType result;
        if(ratio > 0)
        {
           RealType r = 1 + ratio;
           result = cdf(cs, quantile(complement(cs, alpha)) / r) - beta;
        }
        else
        { // ratio <= 0
           RealType r = 1 + ratio;
           result = cdf(complement(cs, quantile(cs, alpha) / r)) - beta;
        }
        return result;
     }
  private:
     RealType alpha;
     RealType beta;
     RealType ratio; // Difference from variance / variance, so fractional.
  };
  
  } // namespace detail
  
  template <class RealType, class Policy>
  RealType chi_squared_distribution<RealType, Policy>::find_degrees_of_freedom(
     RealType difference_from_variance,
     RealType alpha,
     RealType beta,
     RealType variance,
     RealType hint)
  {
     static const char* function = "boost::math::chi_squared_distribution<%1%>::find_degrees_of_freedom(%1%,%1%,%1%,%1%,%1%)";
     // Check for domain errors:
     RealType error_result;
     if(false ==
       detail::check_probability(function, alpha, &error_result, Policy())
       && detail::check_probability(function, beta, &error_result, Policy()))
     { // Either probability is outside 0 to 1.
        return error_result;
     }
  
     if(hint <= 0)
     { // No hint given, so guess df = 1.
        hint = 1;
     }
  
     detail::df_estimator<RealType, Policy> f(alpha, beta, variance, difference_from_variance);
     tools::eps_tolerance<RealType> tol(policies::digits<RealType, Policy>());
     boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();
     std::pair<RealType, RealType> r =
       tools::bracket_and_solve_root(f, hint, RealType(2), false, tol, max_iter, Policy());
     RealType result = r.first + (r.second - r.first) / 2;
     if(max_iter >= policies::get_max_root_iterations<Policy>())
     {
        policies::raise_evaluation_error<RealType>(function, "Unable to locate solution in a reasonable time:"
           " either there is no answer to how many degrees of freedom are required"
           " or the answer is infinite.  Current best guess is %1%", result, Policy());
     }
     return result;
  }
  
  } // 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_MATH_DISTRIBUTIONS_CHI_SQUARED_HPP
  