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130 lines
4.5 KiB
Plaintext
130 lines
4.5 KiB
Plaintext
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[section:inverse_gamma_dist Inverse Gamma Distribution]
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``#include <boost/math/distributions/inverse_gamma.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 inverse_gamma_distribution
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{
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public:
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typedef RealType value_type;
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typedef Policy policy_type;
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inverse_gamma_distribution(RealType shape, RealType scale = 1)
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RealType shape()const;
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RealType scale()const;
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};
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}} // namespaces
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The inverse_gamma distribution is a continuous probability distribution
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of the reciprocal of a variable distributed according to the gamma distribution.
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The inverse_gamma distribution is used in Bayesian statistics.
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See [@http://en.wikipedia.org/wiki/Inverse-gamma_distribution inverse gamma distribution].
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[@http://rss.acs.unt.edu/Rdoc/library/pscl/html/igamma.html R inverse gamma distribution functions].
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[@http://reference.wolfram.com/mathematica/ref/InverseGammaDistribution.html Wolfram inverse gamma distribution].
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See also __gamma_distrib.
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[note
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In spite of potential confusion with the inverse gamma function, this
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distribution *does* provide the typedef:
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``typedef inverse_gamma_distribution<double> gamma;``
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If you want a `double` precision gamma distribution you can use
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``boost::math::inverse_gamma_distribution<>``
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or you can write `inverse_gamma my_ig(2, 3);`]
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For shape parameter [alpha] and scale parameter [beta], it is defined
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by the probability density function (PDF):
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__spaces f(x;[alpha], [beta]) = [beta][super [alpha]] * (1/x) [super [alpha]+1] exp(-[beta]/x) / [Gamma]([alpha])
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and cumulative density function (CDF)
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__spaces F(x;[alpha], [beta]) = [Gamma]([alpha], [beta]/x) / [Gamma]([alpha])
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The following graphs illustrate how the PDF and CDF of the inverse gamma distribution
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varies as the parameters vary:
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[graph inverse_gamma_pdf] [/png or svg]
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[graph inverse_gamma_cdf]
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[h4 Member Functions]
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inverse_gamma_distribution(RealType shape = 1, RealType scale = 1);
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Constructs an inverse gamma distribution with shape [alpha] and scale [beta].
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Requires that the shape and scale parameters are greater than zero, otherwise calls
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__domain_error.
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RealType shape()const;
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Returns the [alpha] shape parameter of this inverse gamma distribution.
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RealType scale()const;
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Returns the [beta] scale parameter of this inverse gamma 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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The domain of the random variate is \[0,+[infin]\].
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[note Unlike some definitions, this implementation supports a random variate
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equal to zero as a special case, returning zero for pdf and cdf.]
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[h4 Accuracy]
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The inverse gamma distribution is implemented in terms of the
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incomplete gamma functions __gamma_p and __gamma_q and their
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inverses __gamma_p_inv and __gamma_q_inv: refer to the accuracy
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data for those functions for more information.
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But in general, inverse_gamma results are accurate to a few epsilon,
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>14 decimal digits accuracy for 64-bit double.
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[h4 Implementation]
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In the following table [alpha] is the shape parameter of the distribution,
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[alpha][space] is its scale parameter, /x/ is the random variate, /p/ is the probability
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and /q = 1-p/.
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[table
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[[Function][Implementation Notes]]
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[[pdf][Using the relation: pdf = __gamma_p_derivative([alpha], [beta]/ x, [beta]) / x * x ]]
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[[cdf][Using the relation: p = __gamma_q([alpha], [beta] / x) ]]
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[[cdf complement][Using the relation: q = __gamma_p([alpha], [beta] / x) ]]
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[[quantile][Using the relation: x = [beta][space]/ __gamma_q_inv([alpha], p) ]]
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[[quantile from the complement][Using the relation: x = [alpha][space]/ __gamma_p_inv([alpha], q) ]]
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[[mode][[beta] / ([alpha] + 1) ]]
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[[median][no analytic equation is known, but is evaluated as quantile(0.5)]]
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[[mean][[beta] / ([alpha] - 1) for [alpha] > 1, else a __domain_error]]
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[[variance][([beta] * [beta]) / (([alpha] - 1) * ([alpha] - 1) * ([alpha] - 2)) for [alpha] >2, else a __domain_error]]
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[[skewness][4 * sqrt ([alpha] -2) / ([alpha] -3) for [alpha] >3, else a __domain_error]]
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[[kurtosis_excess][(30 * [alpha] - 66) / (([alpha]-3)*([alpha] - 4)) for [alpha] >4, else a __domain_error]]
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] [/table]
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[endsect][/section:inverse_gamma_dist Inverse Gamma Distribution]
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[/
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Copyright 2010 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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