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