boost/math/distributions/pareto.hpp
// 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 l_scale = 1, RealType l_shape = 1)
: m_scale(l_scale), m_shape(l_shape)
{ // Constructor.
RealType result = 0;
detail::check_pareto("boost::math::pareto_distribution<%1%>::pareto_distribution", l_scale, l_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.);
#ifdef __cpp_deduction_guides
template <class RealType>
pareto_distribution(RealType)->pareto_distribution<typename boost::math::tools::promote_args<RealType>::type>;
template <class RealType>
pareto_distribution(RealType,RealType)->pareto_distribution<typename boost::math::tools::promote_args<RealType>::type>;
#endif
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 = 0;
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 = 0;
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 = 0;
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 policies::raise_overflow_error<RealType>(function, 0, Policy()); // 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 = 0;
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 = 0;
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 policies::raise_overflow_error<RealType>(function, 0, Policy()); // 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 = 0;
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 = 0;
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 = 0;
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 = 0;
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 = 0;
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 = 0;
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
template <class RealType, class Policy>
inline RealType entropy(const pareto_distribution<RealType, Policy>& dist)
{
using std::log;
RealType xm = dist.scale();
RealType alpha = dist.shape();
return log(xm/alpha) + 1 + 1/alpha;
}
} // 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