Chris@16: //  Copyright John Maddock 2007.
Chris@16: //  Copyright Paul A. Bristow 2007, 2009
Chris@16: //  Use, modification and distribution are subject to the
Chris@16: //  Boost Software License, Version 1.0. (See accompanying file
Chris@16: //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
Chris@16: 
Chris@16: #ifndef BOOST_STATS_PARETO_HPP
Chris@16: #define BOOST_STATS_PARETO_HPP
Chris@16: 
Chris@16: // http://en.wikipedia.org/wiki/Pareto_distribution
Chris@16: // http://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm
Chris@16: // Also:
Chris@16: // Weisstein, Eric W. "Pareto Distribution."
Chris@16: // From MathWorld--A Wolfram Web Resource.
Chris@16: // http://mathworld.wolfram.com/ParetoDistribution.html
Chris@16: // Handbook of Statistical Distributions with Applications, K Krishnamoorthy, ISBN 1-58488-635-8, Chapter 23, pp 257 - 267.
Chris@16: // Caution KK's a and b are the reverse of Mathworld!
Chris@16: 
Chris@16: #include <boost/math/distributions/fwd.hpp>
Chris@16: #include <boost/math/distributions/complement.hpp>
Chris@16: #include <boost/math/distributions/detail/common_error_handling.hpp>
Chris@16: #include <boost/math/special_functions/powm1.hpp>
Chris@16: 
Chris@16: #include <utility> // for BOOST_CURRENT_VALUE?
Chris@16: 
Chris@16: namespace boost
Chris@16: {
Chris@16:   namespace math
Chris@16:   {
Chris@16:     namespace detail
Chris@16:     { // Parameter checking.
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline bool check_pareto_scale(
Chris@16:         const char* function,
Chris@16:         RealType scale,
Chris@16:         RealType* result, const Policy& pol)
Chris@16:       {
Chris@16:         if((boost::math::isfinite)(scale))
Chris@16:         { // any > 0 finite value is OK.
Chris@16:           if (scale > 0)
Chris@16:           {
Chris@16:             return true;
Chris@16:           }
Chris@16:           else
Chris@16:           {
Chris@16:             *result = policies::raise_domain_error<RealType>(
Chris@16:               function,
Chris@16:               "Scale parameter is %1%, but must be > 0!", scale, pol);
Chris@16:             return false;
Chris@16:           }
Chris@16:         }
Chris@16:         else
Chris@16:         { // Not finite.
Chris@16:           *result = policies::raise_domain_error<RealType>(
Chris@16:             function,
Chris@16:             "Scale parameter is %1%, but must be finite!", scale, pol);
Chris@16:           return false;
Chris@16:         }
Chris@16:       } // bool check_pareto_scale
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline bool check_pareto_shape(
Chris@16:         const char* function,
Chris@16:         RealType shape,
Chris@16:         RealType* result, const Policy& pol)
Chris@16:       {
Chris@16:         if((boost::math::isfinite)(shape))
Chris@16:         { // Any finite value > 0 is OK.
Chris@16:           if (shape > 0)
Chris@16:           {
Chris@16:             return true;
Chris@16:           }
Chris@16:           else
Chris@16:           {
Chris@16:             *result = policies::raise_domain_error<RealType>(
Chris@16:               function,
Chris@16:               "Shape parameter is %1%, but must be > 0!", shape, pol);
Chris@16:             return false;
Chris@16:           }
Chris@16:         }
Chris@16:         else
Chris@16:         { // Not finite.
Chris@16:           *result = policies::raise_domain_error<RealType>(
Chris@16:             function,
Chris@16:             "Shape parameter is %1%, but must be finite!", shape, pol);
Chris@16:           return false;
Chris@16:         }
Chris@16:       } // bool check_pareto_shape(
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline bool check_pareto_x(
Chris@16:         const char* function,
Chris@16:         RealType const& x,
Chris@16:         RealType* result, const Policy& pol)
Chris@16:       {
Chris@16:         if((boost::math::isfinite)(x))
Chris@16:         { //
Chris@16:           if (x > 0)
Chris@16:           {
Chris@16:             return true;
Chris@16:           }
Chris@16:           else
Chris@16:           {
Chris@16:             *result = policies::raise_domain_error<RealType>(
Chris@16:               function,
Chris@16:               "x parameter is %1%, but must be > 0 !", x, pol);
Chris@16:             return false;
Chris@16:           }
Chris@16:         }
Chris@16:         else
Chris@16:         { // Not finite..
Chris@16:           *result = policies::raise_domain_error<RealType>(
Chris@16:             function,
Chris@16:             "x parameter is %1%, but must be finite!", x, pol);
Chris@16:           return false;
Chris@16:         }
Chris@16:       } // bool check_pareto_x
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline bool check_pareto( // distribution parameters.
Chris@16:         const char* function,
Chris@16:         RealType scale,
Chris@16:         RealType shape,
Chris@16:         RealType* result, const Policy& pol)
Chris@16:       {
Chris@16:         return check_pareto_scale(function, scale, result, pol)
Chris@16:            && check_pareto_shape(function, shape, result, pol);
Chris@16:       } // bool check_pareto(
Chris@16: 
Chris@16:     } // namespace detail
Chris@16: 
Chris@16:     template <class RealType = double, class Policy = policies::policy<> >
Chris@16:     class pareto_distribution
Chris@16:     {
Chris@16:     public:
Chris@16:       typedef RealType value_type;
Chris@16:       typedef Policy policy_type;
Chris@16: 
Chris@16:       pareto_distribution(RealType l_scale = 1, RealType l_shape = 1)
Chris@16:         : m_scale(l_scale), m_shape(l_shape)
Chris@16:       { // Constructor.
Chris@16:         RealType result = 0;
Chris@16:         detail::check_pareto("boost::math::pareto_distribution<%1%>::pareto_distribution", l_scale, l_shape, &result, Policy());
Chris@16:       }
Chris@16: 
Chris@16:       RealType scale()const
Chris@16:       { // AKA Xm and Wolfram b and beta
Chris@16:         return m_scale;
Chris@16:       }
Chris@16: 
Chris@16:       RealType shape()const
Chris@16:       { // AKA k and Wolfram a and alpha
Chris@16:         return m_shape;
Chris@16:       }
Chris@16:     private:
Chris@16:       // Data members:
Chris@16:       RealType m_scale;  // distribution scale (xm) or beta
Chris@16:       RealType m_shape;  // distribution shape (k) or alpha
Chris@16:     };
Chris@16: 
Chris@16:     typedef pareto_distribution<double> pareto; // Convenience to allow pareto(2., 3.);
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline const std::pair<RealType, RealType> range(const pareto_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>()); // scale zero to + infinity.
Chris@16:     } // range
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline const std::pair<RealType, RealType> support(const pareto_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:       using boost::math::tools::max_value;
Chris@16:       return std::pair<RealType, RealType>(dist.scale(), max_value<RealType>() ); // scale to + infinity.
Chris@16:     } // support
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType pdf(const pareto_distribution<RealType, Policy>& dist, const RealType& x)
Chris@16:     {
Chris@16:       BOOST_MATH_STD_USING  // for ADL of std function pow.
Chris@16:       static const char* function = "boost::math::pdf(const pareto_distribution<%1%>&, %1%)";
Chris@16:       RealType scale = dist.scale();
Chris@16:       RealType shape = dist.shape();
Chris@16:       RealType result = 0;
Chris@16:       if(false == (detail::check_pareto_x(function, x, &result, Policy())
Chris@16:          && detail::check_pareto(function, scale, shape, &result, Policy())))
Chris@16:          return result;
Chris@16:       if (x < scale)
Chris@16:       { // regardless of shape, pdf is zero (or should be disallow x < scale and throw an exception?).
Chris@16:         return 0;
Chris@16:       }
Chris@16:       result = shape * pow(scale, shape) / pow(x, shape+1);
Chris@16:       return result;
Chris@16:     } // pdf
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType cdf(const pareto_distribution<RealType, Policy>& dist, const RealType& x)
Chris@16:     {
Chris@16:       BOOST_MATH_STD_USING  // for ADL of std function pow.
Chris@16:       static const char* function = "boost::math::cdf(const pareto_distribution<%1%>&, %1%)";
Chris@16:       RealType scale = dist.scale();
Chris@16:       RealType shape = dist.shape();
Chris@16:       RealType result = 0;
Chris@16: 
Chris@16:       if(false == (detail::check_pareto_x(function, x, &result, Policy())
Chris@16:          && detail::check_pareto(function, scale, shape, &result, Policy())))
Chris@16:          return result;
Chris@16: 
Chris@16:       if (x <= scale)
Chris@16:       { // regardless of shape, cdf is zero.
Chris@16:         return 0;
Chris@16:       }
Chris@16: 
Chris@16:       // result = RealType(1) - pow((scale / x), shape);
Chris@16:       result = -boost::math::powm1(scale/x, shape, Policy()); // should be more accurate.
Chris@16:       return result;
Chris@16:     } // cdf
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType quantile(const pareto_distribution<RealType, Policy>& dist, const RealType& p)
Chris@16:     {
Chris@16:       BOOST_MATH_STD_USING  // for ADL of std function pow.
Chris@16:       static const char* function = "boost::math::quantile(const pareto_distribution<%1%>&, %1%)";
Chris@16:       RealType result = 0;
Chris@16:       RealType scale = dist.scale();
Chris@16:       RealType shape = dist.shape();
Chris@16:       if(false == (detail::check_probability(function, p, &result, Policy())
Chris@16:            && detail::check_pareto(function, scale, shape, &result, Policy())))
Chris@16:       {
Chris@16:         return result;
Chris@16:       }
Chris@16:       if (p == 0)
Chris@16:       {
Chris@16:         return scale; // x must be scale (or less).
Chris@16:       }
Chris@16:       if (p == 1)
Chris@16:       {
Chris@101:         return policies::raise_overflow_error<RealType>(function, 0, Policy()); // x = + infinity.
Chris@16:       }
Chris@16:       result = scale /
Chris@16:         (pow((1 - p), 1 / shape));
Chris@16:       // K. Krishnamoorthy,  ISBN 1-58488-635-8 eq 23.1.3
Chris@16:       return result;
Chris@16:     } // quantile
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType cdf(const complemented2_type<pareto_distribution<RealType, Policy>, RealType>& c)
Chris@16:     {
Chris@16:        BOOST_MATH_STD_USING  // for ADL of std function pow.
Chris@16:        static const char* function = "boost::math::cdf(const pareto_distribution<%1%>&, %1%)";
Chris@16:        RealType result = 0;
Chris@16:        RealType x = c.param;
Chris@16:        RealType scale = c.dist.scale();
Chris@16:        RealType shape = c.dist.shape();
Chris@16:        if(false == (detail::check_pareto_x(function, x, &result, Policy())
Chris@16:            && detail::check_pareto(function, scale, shape, &result, Policy())))
Chris@16:          return result;
Chris@16: 
Chris@16:        if (x <= scale)
Chris@16:        { // regardless of shape, cdf is zero, and complement is unity.
Chris@16:          return 1;
Chris@16:        }
Chris@16:        result = pow((scale/x), shape);
Chris@16: 
Chris@16:        return result;
Chris@16:     } // cdf complement
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType quantile(const complemented2_type<pareto_distribution<RealType, Policy>, RealType>& c)
Chris@16:     {
Chris@16:       BOOST_MATH_STD_USING  // for ADL of std function pow.
Chris@16:       static const char* function = "boost::math::quantile(const pareto_distribution<%1%>&, %1%)";
Chris@16:       RealType result = 0;
Chris@16:       RealType q = c.param;
Chris@16:       RealType scale = c.dist.scale();
Chris@16:       RealType shape = c.dist.shape();
Chris@16:       if(false == (detail::check_probability(function, q, &result, Policy())
Chris@16:            && detail::check_pareto(function, scale, shape, &result, Policy())))
Chris@16:       {
Chris@16:         return result;
Chris@16:       }
Chris@16:       if (q == 1)
Chris@16:       {
Chris@16:         return scale; // x must be scale (or less).
Chris@16:       }
Chris@16:       if (q == 0)
Chris@16:       {
Chris@101:          return policies::raise_overflow_error<RealType>(function, 0, Policy()); // x = + infinity.
Chris@16:       }
Chris@16:       result = scale / (pow(q, 1 / shape));
Chris@16:       // K. Krishnamoorthy,  ISBN 1-58488-635-8 eq 23.1.3
Chris@16:       return result;
Chris@16:     } // quantile complement
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType mean(const pareto_distribution<RealType, Policy>& dist)
Chris@16:     {
Chris@16:       RealType result = 0;
Chris@16:       static const char* function = "boost::math::mean(const pareto_distribution<%1%>&, %1%)";
Chris@16:       if(false == detail::check_pareto(function, dist.scale(), dist.shape(), &result, Policy()))
Chris@16:       {
Chris@16:         return result;
Chris@16:       }
Chris@16:       if (dist.shape() > RealType(1))
Chris@16:       {
Chris@16:         return dist.shape() * dist.scale() / (dist.shape() - 1);
Chris@16:       }
Chris@16:       else
Chris@16:       {
Chris@16:         using boost::math::tools::max_value;
Chris@16:         return max_value<RealType>(); // +infinity.
Chris@16:       }
Chris@16:     } // mean
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType mode(const pareto_distribution<RealType, Policy>& dist)
Chris@16:     {
Chris@16:       return dist.scale();
Chris@16:     } // mode
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType median(const pareto_distribution<RealType, Policy>& dist)
Chris@16:     {
Chris@16:       RealType result = 0;
Chris@16:       static const char* function = "boost::math::median(const pareto_distribution<%1%>&, %1%)";
Chris@16:       if(false == detail::check_pareto(function, dist.scale(), dist.shape(), &result, Policy()))
Chris@16:       {
Chris@16:         return result;
Chris@16:       }
Chris@16:       BOOST_MATH_STD_USING
Chris@16:       return dist.scale() * pow(RealType(2), (1/dist.shape()));
Chris@16:     } // median
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType variance(const pareto_distribution<RealType, Policy>& dist)
Chris@16:     {
Chris@16:       RealType result = 0;
Chris@16:       RealType scale = dist.scale();
Chris@16:       RealType shape = dist.shape();
Chris@16:       static const char* function = "boost::math::variance(const pareto_distribution<%1%>&, %1%)";
Chris@16:       if(false == detail::check_pareto(function, scale, shape, &result, Policy()))
Chris@16:       {
Chris@16:         return result;
Chris@16:       }
Chris@16:       if (shape > 2)
Chris@16:       {
Chris@16:         result = (scale * scale * shape) /
Chris@16:          ((shape - 1) *  (shape - 1) * (shape - 2));
Chris@16:       }
Chris@16:       else
Chris@16:       {
Chris@16:         result = policies::raise_domain_error<RealType>(
Chris@16:           function,
Chris@16:           "variance is undefined for shape <= 2, but got %1%.", dist.shape(), Policy());
Chris@16:       }
Chris@16:       return result;
Chris@16:     } // variance
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType skewness(const pareto_distribution<RealType, Policy>& dist)
Chris@16:     {
Chris@16:       BOOST_MATH_STD_USING
Chris@16:       RealType result = 0;
Chris@16:       RealType shape = dist.shape();
Chris@16:       static const char* function = "boost::math::pdf(const pareto_distribution<%1%>&, %1%)";
Chris@16:       if(false == detail::check_pareto(function, dist.scale(), shape, &result, Policy()))
Chris@16:       {
Chris@16:         return result;
Chris@16:       }
Chris@16:       if (shape > 3)
Chris@16:       {
Chris@16:         result = sqrt((shape - 2) / shape) *
Chris@16:           2 * (shape + 1) /
Chris@16:           (shape - 3);
Chris@16:       }
Chris@16:       else
Chris@16:       {
Chris@16:         result = policies::raise_domain_error<RealType>(
Chris@16:           function,
Chris@16:           "skewness is undefined for shape <= 3, but got %1%.", dist.shape(), Policy());
Chris@16:       }
Chris@16:       return result;
Chris@16:     } // skewness
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType kurtosis(const pareto_distribution<RealType, Policy>& dist)
Chris@16:     {
Chris@16:       RealType result = 0;
Chris@16:       RealType shape = dist.shape();
Chris@16:       static const char* function = "boost::math::pdf(const pareto_distribution<%1%>&, %1%)";
Chris@16:       if(false == detail::check_pareto(function, dist.scale(), shape, &result, Policy()))
Chris@16:       {
Chris@16:         return result;
Chris@16:       }
Chris@16:       if (shape > 4)
Chris@16:       {
Chris@16:         result = 3 * ((shape - 2) * (3 * shape * shape + shape + 2)) /
Chris@16:           (shape * (shape - 3) * (shape - 4));
Chris@16:       }
Chris@16:       else
Chris@16:       {
Chris@16:         result = policies::raise_domain_error<RealType>(
Chris@16:           function,
Chris@16:           "kurtosis_excess is undefined for shape <= 4, but got %1%.", shape, Policy());
Chris@16:       }
Chris@16:       return result;
Chris@16:     } // kurtosis
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType kurtosis_excess(const pareto_distribution<RealType, Policy>& dist)
Chris@16:     {
Chris@16:       RealType result = 0;
Chris@16:       RealType shape = dist.shape();
Chris@16:       static const char* function = "boost::math::pdf(const pareto_distribution<%1%>&, %1%)";
Chris@16:       if(false == detail::check_pareto(function, dist.scale(), shape, &result, Policy()))
Chris@16:       {
Chris@16:         return result;
Chris@16:       }
Chris@16:       if (shape > 4)
Chris@16:       {
Chris@16:         result = 6 * ((shape * shape * shape) + (shape * shape) - 6 * shape - 2) /
Chris@16:           (shape * (shape - 3) * (shape - 4));
Chris@16:       }
Chris@16:       else
Chris@16:       {
Chris@16:         result = policies::raise_domain_error<RealType>(
Chris@16:           function,
Chris@16:           "kurtosis_excess is undefined for shape <= 4, but got %1%.", dist.shape(), Policy());
Chris@16:       }
Chris@16:       return result;
Chris@16:     } // kurtosis_excess
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_STATS_PARETO_HPP
Chris@16: 
Chris@16: