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126 lines
4.8 KiB
Plaintext
126 lines
4.8 KiB
Plaintext
[section:pareto Pareto Distribution]
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``#include <boost/math/distributions/pareto.hpp>``
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namespace boost{ namespace math{
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template <class RealType = double,
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class ``__Policy`` = ``__policy_class`` >
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class pareto_distribution;
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typedef pareto_distribution<> pareto;
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template <class RealType, class ``__Policy``>
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class pareto_distribution
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{
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public:
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typedef RealType value_type;
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// Constructor:
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BOOST_MATH_GPU_ENABLED pareto_distribution(RealType scale = 1, RealType shape = 1)
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// Accessors:
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BOOST_MATH_GPU_ENABLED RealType scale()const;
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BOOST_MATH_GPU_ENABLED RealType shape()const;
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};
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}} // namespaces
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The [@http://en.wikipedia.org/wiki/pareto_distribution Pareto distribution]
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is a continuous distribution with the
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[@http://en.wikipedia.org/wiki/Probability_density_function probability density function (pdf)]:
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[expression f(x; [alpha], [beta]) = [alpha][beta][super [alpha]] / x[super [alpha]+ 1]]
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For shape parameter [alpha] > 0, and scale parameter [beta] > 0.
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If x < [beta], the pdf is zero.
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The [@http://mathworld.wolfram.com/ParetoDistribution.html Pareto distribution]
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often describes the larger compared to the smaller.
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A classic example is that 80% of the wealth is owned by 20% of the population.
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The following graph illustrates how the PDF varies with the scale parameter [beta]:
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[graph pareto_pdf1]
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And this graph illustrates how the PDF varies with the shape parameter [alpha]:
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[graph pareto_pdf2]
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[h4 Related distributions]
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[h4 Member Functions]
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BOOST_MATH_GPU_ENABLED pareto_distribution(RealType scale = 1, RealType shape = 1);
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Constructs a [@http://en.wikipedia.org/wiki/pareto_distribution
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pareto distribution] with shape /shape/ and scale /scale/.
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Requires that the /shape/ and /scale/ parameters are both greater than zero,
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otherwise calls __domain_error.
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BOOST_MATH_GPU_ENABLED RealType scale()const;
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Returns the /scale/ parameter of this distribution.
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BOOST_MATH_GPU_ENABLED RealType shape()const;
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Returns the /shape/ parameter of this distribution.
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[h4 Non-member Accessors]
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All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all
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distributions are supported: __usual_accessors.
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For this distribution all non-member accessor functions are marked with `BOOST_MATH_GPU_ENABLED` and can
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be run on both host and device.
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The supported domain of the random variable is \[scale, [infin]\].
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In this distribution the implementation of `logcdf` is specialized
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to improve numerical accuracy.
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[h4 Accuracy]
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The Pareto distribution is implemented in terms of the
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standard library `exp` functions plus __expm1
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and so should have very small errors, usually only a few epsilon.
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If probability is near to unity (or the complement of a probability near zero) see also __why_complements.
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[h4 Implementation]
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In the following table [alpha] is the shape parameter of the distribution, and
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[beta] is its scale parameter, /x/ is the random variate, /p/ is the probability
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and its complement /q = 1-p/.
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[table
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[[Function][Implementation Notes]]
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[[pdf][Using the relation: pdf p = [alpha][beta][super [alpha]]/x[super [alpha] +1] ]]
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[[cdf][Using the relation: cdf p = 1 - ([beta] / x)[super [alpha]] ]]
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[[logcdf][log(cdf) = log1p(-pow([beta]/x, [alpha])) ]]
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[[cdf complement][Using the relation: q = 1 - p = -([beta] / x)[super [alpha]] ]]
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[[quantile][Using the relation: x = [beta] / (1 - p)[super 1/[alpha]] ]]
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[[quantile from the complement][Using the relation: x = [beta] / (q)[super 1/[alpha]] ]]
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[[mean][[alpha][beta] / ([beta] - 1) ]]
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[[variance][[beta][alpha][super 2] / ([beta] - 1)[super 2] ([beta] - 2) ]]
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[[mode][[alpha]]]
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[[skewness][Refer to [@http://mathworld.wolfram.com/ParetoDistribution.html Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.] ]]
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[[kurtosis][Refer to [@http://mathworld.wolfram.com/ParetoDistribution.html Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.] ]]
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[[kurtosis excess][Refer to [@http://mathworld.wolfram.com/ParetoDistribution.html Weisstein, Eric W. "pareto Distribution." From MathWorld--A Wolfram Web Resource.] ]]
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]
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[h4 References]
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* [@http://en.wikipedia.org/wiki/pareto_distribution Pareto Distribution]
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* [@http://mathworld.wolfram.com/paretoDistribution.html Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.]
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* Handbook of Statistical Distributions with Applications, K Krishnamoorthy, ISBN 1-58488-635-8, Chapter 23, pp 257 - 267.
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(Note the meaning of a and b is reversed in Wolfram and Krishnamoorthy).
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[endsect] [/section:pareto pareto]
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[/
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Copyright 2006, 2009 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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