| // 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.
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| template <class RealType, class Policy>
|
| inline bool check_pareto_scale(
|
| const char* function,
|
| RealType scale,
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| 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(
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| const char* function,
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| RealType shape,
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| 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(
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| const char* function,
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| RealType const& x,
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| 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);
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| return false;
|
| }
|
| } // bool check_pareto_x
|
|
|
| template <class RealType, class Policy>
|
| inline bool check_pareto( // distribution parameters.
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| const char* function,
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| RealType scale,
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| RealType shape,
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| RealType* result, const Policy& pol)
|
| {
|
| return check_pareto_scale(function, scale, result, pol)
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| && check_pareto_shape(function, shape, result, pol);
|
| } // bool check_pareto(
|
|
|
| } // namespace detail
|
|
|
| template <class RealType = double, class Policy = policies::policy<> >
|
| class pareto_distribution
|
| {
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| public:
|
| typedef RealType value_type;
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| typedef Policy policy_type;
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|
|
| pareto_distribution(RealType scale = 1, RealType shape = 1)
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| : m_scale(scale), m_shape(shape)
|
| { // Constructor.
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| RealType result;
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| detail::check_pareto("boost::math::pareto_distribution<%1%>::pareto_distribution", scale, shape, &result, Policy());
|
| }
|
|
|
| RealType scale()const
|
| { // AKA Xm and Wolfram b and beta
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| return m_scale;
|
| }
|
|
|
| RealType shape()const
|
| { // AKA k and Wolfram a and alpha
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| 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*/)
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| { // Range of permissible values for random variable x.
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| using boost::math::tools::max_value;
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| 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.
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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;
|
| 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();
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| 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();
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| 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();
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| RealType shape = dist.shape();
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| 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();
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| 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();
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| RealType shape = c.dist.shape();
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| 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
|
|
|
|
|
