[section:pareto Pareto Distribution]


``#include <boost/math/distributions/pareto.hpp>``

   namespace boost{ namespace math{

   template <class RealType = double,
             class ``__Policy``   = ``__policy_class`` >
   class pareto_distribution;

   typedef pareto_distribution<> pareto;

   template <class RealType, class ``__Policy``>
   class pareto_distribution
   {
   public:
      typedef RealType value_type;
      // Constructor:
      BOOST_MATH_GPU_ENABLED pareto_distribution(RealType scale = 1, RealType shape = 1)
      // Accessors:
      BOOST_MATH_GPU_ENABLED RealType scale()const;
      BOOST_MATH_GPU_ENABLED RealType shape()const;
   };

   }} // namespaces

The [@http://en.wikipedia.org/wiki/pareto_distribution Pareto distribution]
is a continuous distribution with the
[@http://en.wikipedia.org/wiki/Probability_density_function probability density function (pdf)]:

[expression f(x; [alpha], [beta]) = [alpha][beta][super [alpha]] / x[super [alpha]+ 1]]

For shape parameter [alpha] > 0, and scale parameter [beta] > 0.
If x < [beta], the pdf is zero.

The [@http://mathworld.wolfram.com/ParetoDistribution.html Pareto distribution]
often describes the larger compared to the smaller.
A classic example is that 80% of the wealth is owned by 20% of the population.

The following graph illustrates how the PDF varies with the scale parameter [beta]:

[graph pareto_pdf1]

And this graph illustrates how the PDF varies with the shape parameter [alpha]:

[graph pareto_pdf2]

[h4 Related distributions]

[h4 Member Functions]

   BOOST_MATH_GPU_ENABLED pareto_distribution(RealType scale = 1, RealType shape = 1);

Constructs a [@http://en.wikipedia.org/wiki/pareto_distribution
pareto distribution] with shape /shape/ and scale /scale/.

Requires that the /shape/ and /scale/ parameters are both greater than zero,
otherwise calls __domain_error.

   BOOST_MATH_GPU_ENABLED RealType scale()const;

Returns the /scale/ parameter of this distribution.

   BOOST_MATH_GPU_ENABLED RealType shape()const;

Returns the /shape/ parameter of this distribution.

[h4 Non-member Accessors]

All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all
distributions are supported: __usual_accessors.
For this distribution all non-member accessor functions are marked with `BOOST_MATH_GPU_ENABLED` and can
be run on both host and device.

The supported domain of the random variable is \[scale, [infin]\].

In this distribution the implementation of `logcdf` is specialized
to improve numerical accuracy.

[h4 Accuracy]

The Pareto distribution is implemented in terms of the
standard library `exp` functions plus __expm1
and so should have very small errors, usually only a few epsilon.

If probability is near to unity (or the complement of a probability near zero) see also __why_complements.

[h4 Implementation]

In the following table [alpha] is the shape parameter of the distribution, and
[beta] is its scale parameter, /x/ is the random variate, /p/ is the probability
and its complement /q = 1-p/.

[table
[[Function][Implementation Notes]]
[[pdf][Using the relation: pdf p = [alpha][beta][super [alpha]]/x[super [alpha] +1] ]]
[[cdf][Using the relation: cdf p = 1 - ([beta] / x)[super [alpha]] ]]
[[logcdf][log(cdf) = log1p(-pow([beta]/x, [alpha])) ]]
[[cdf complement][Using the relation: q = 1 - p = -([beta] / x)[super [alpha]] ]]
[[quantile][Using the relation: x = [beta] / (1 - p)[super 1/[alpha]] ]]
[[quantile from the complement][Using the relation: x =  [beta] / (q)[super 1/[alpha]] ]]
[[mean][[alpha][beta] / ([beta] - 1) ]]
[[variance][[beta][alpha][super 2] / ([beta] - 1)[super 2] ([beta] - 2) ]]
[[mode][[alpha]]]
[[skewness][Refer to [@http://mathworld.wolfram.com/ParetoDistribution.html Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.] ]]
[[kurtosis][Refer to [@http://mathworld.wolfram.com/ParetoDistribution.html Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.] ]]
[[kurtosis excess][Refer to [@http://mathworld.wolfram.com/ParetoDistribution.html Weisstein, Eric W. "pareto Distribution." From MathWorld--A Wolfram Web Resource.] ]]
]

[h4 References]
* [@http://en.wikipedia.org/wiki/pareto_distribution Pareto Distribution]
* [@http://mathworld.wolfram.com/paretoDistribution.html Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.]
* Handbook of Statistical Distributions with Applications, K Krishnamoorthy, ISBN 1-58488-635-8, Chapter 23, pp 257 - 267.
(Note the meaning of a and b is reversed in Wolfram and Krishnamoorthy).

[endsect] [/section:pareto pareto]

[/
  Copyright 2006, 2009 John Maddock and Paul A. Bristow.
  Distributed under 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).
]

