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176 lines
7.9 KiB
Plaintext
176 lines
7.9 KiB
Plaintext
[section:inverse_chi_squared_dist Inverse Chi Squared Distribution]
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``#include <boost/math/distributions/inverse_chi_squared.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_chi_squared_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_chi_squared_distribution(RealType df = 1); // Not explicitly scaled, default 1/df.
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inverse_chi_squared_distribution(RealType df, RealType scale = 1/df); // Scaled.
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RealType degrees_of_freedom()const; // Default 1.
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RealType scale()const; // Optional scale [xi] (variance), default 1/degrees_of_freedom.
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};
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}} // namespace boost // namespace math
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The inverse chi squared distribution is a continuous probability distribution
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of the *reciprocal* of a variable distributed according to the chi squared distribution.
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The sources below give confusingly different formulae
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using different symbols for the distribution pdf,
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but they are all the same, or related by a change of variable, or choice of scale.
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Two constructors are available to implement both the scaled and (implicitly) unscaled versions.
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The main version has an explicit scale parameter which implements the
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[@http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution scaled inverse chi_squared distribution].
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A second version has an implicit scale = 1/degrees of freedom and gives the 1st definition in the
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[@http://en.wikipedia.org/wiki/Inverse-chi-square_distribution Wikipedia inverse chi_squared distribution].
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The 2nd Wikipedia inverse chi_squared distribution definition can be implemented
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by explicitly specifying a scale = 1.
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Both definitions are also available in Wolfram Mathematica and in __R (geoR) with default scale = 1/degrees of freedom.
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See
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* Inverse chi_squared distribution [@http://en.wikipedia.org/wiki/Inverse-chi-square_distribution]
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* Scaled inverse chi_squared distribution[@http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution]
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* R inverse chi_squared distribution functions [@http://hosho.ees.hokudai.ac.jp/~kubo/Rdoc/library/geoR/html/InvChisquare.html R ]
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* Inverse chi_squared distribution functions [@http://mathworld.wolfram.com/InverseChi-SquaredDistribution.html Weisstein, Eric W. "Inverse Chi-Squared Distribution." From MathWorld--A Wolfram Web Resource.]
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* Inverse chi_squared distribution reference [@http://reference.wolfram.com/mathematica/ref/InverseChiSquareDistribution.html Weisstein, Eric W. "Inverse Chi-Squared Distribution reference." From Wolfram Mathematica.]
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The inverse_chi_squared distribution is used in
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[@http://en.wikipedia.org/wiki/Bayesian_statistics Bayesian statistics]:
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the scaled inverse chi-square is conjugate prior for the normal distribution
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with known mean, model parameter [sigma][pow2] (variance).
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See [@http://en.wikipedia.org/wiki/Conjugate_prior conjugate priors including a table of distributions and their priors.]
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See also __inverse_gamma_distrib and __chi_squared_distrib.
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The inverse_chi_squared distribution is a special case of a inverse_gamma distribution
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with [nu] (degrees_of_freedom) shape ([alpha]) and scale ([beta]) where
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__spaces [alpha]= [nu] /2 and [beta] = [frac12].
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[note This distribution *does* provide the typedef:
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``typedef inverse_chi_squared_distribution<double> inverse_chi_squared;``
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If you want a `double` precision inverse_chi_squared distribution you can use
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``boost::math::inverse_chi_squared_distribution<>``
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or you can write `inverse_chi_squared my_invchisqr(2, 3);`]
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For degrees of freedom parameter [nu],
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the (*unscaled*) inverse chi_squared distribution is defined by the probability density function (PDF):
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__spaces f(x;[nu]) = 2[super -[nu]/2] x[super -[nu]/2-1] e[super -1/2x] / [Gamma]([nu]/2)
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and Cumulative Density Function (CDF)
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__spaces F(x;[nu]) = [Gamma]([nu]/2, 1/2x) / [Gamma]([nu]/2)
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For degrees of freedom parameter [nu] and scale parameter [xi],
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the *scaled* inverse chi_squared distribution is defined by the probability density function (PDF):
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__spaces f(x;[nu], [xi]) = ([xi][nu]/2)[super [nu]/2] e[super -[nu][xi]/2x] x[super -1-[nu]/2] / [Gamma]([nu]/2)
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and Cumulative Density Function (CDF)
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__spaces F(x;[nu], [xi]) = [Gamma]([nu]/2, [nu][xi]/2x) / [Gamma]([nu]/2)
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The following graphs illustrate how the PDF and CDF of the inverse chi_squared distribution
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varies for a few values of parameters [nu] and [xi]:
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[graph inverse_chi_squared_pdf] [/.png or .svg]
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[graph inverse_chi_squared_cdf]
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[h4 Member Functions]
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inverse_chi_squared_distribution(RealType df = 1); // Implicitly scaled 1/df.
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inverse_chi_squared_distribution(RealType df = 1, RealType scale); // Explicitly scaled.
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Constructs an inverse chi_squared distribution with [nu] degrees of freedom ['df],
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and scale ['scale] with default value 1\/df.
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Requires that the degrees of freedom [nu] parameter is greater than zero, otherwise calls
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__domain_error.
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RealType degrees_of_freedom()const;
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Returns the degrees_of_freedom [nu] parameter of this distribution.
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RealType scale()const;
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Returns the scale [xi] 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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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 both 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 like the __inverse_gamma_distrib that use
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__gamma_p and __gamma_q and their inverses __gamma_p_inv and __gamma_q_inv:
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refer to the accuracy data for those functions for more information.
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But in general, gamma (and thus inverse gamma) results are often accurate to a few epsilon,
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>14 decimal digits accuracy for 64-bit double.
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unless iteration is involved, as for the estimation of degrees of freedom.
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[h4 Implementation]
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In the following table [nu] is the degrees of freedom parameter and
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[xi] is the scale parameter of the distribution,
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/x/ is the random variate, /p/ is the probability and /q = 1-p/ its complement.
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Parameters [alpha] for shape and [beta] for scale
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are used for the inverse gamma function: [alpha] = [nu]/2 and [beta] = [nu] * [xi]/2.
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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][[nu] * [xi] / ([nu] + 2) ]]
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[[median][no closed form analytic equation is known, but is evaluated as quantile(0.5)]]
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[[mean][[nu][xi] / ([nu] - 2) for [nu] > 2, else a __domain_error]]
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[[variance][2 [nu][pow2] [xi][pow2] / (([nu] -2)[pow2] ([nu] -4)) for [nu] >4, else a __domain_error]]
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[[skewness][4 [sqrt]2 [sqrt]([nu]-4) /([nu]-6) for [nu] >6, else a __domain_error ]]
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[[kurtosis_excess][12 * (5[nu] - 22) / (([nu] - 6) * ([nu] - 8)) for [nu] >8, else a __domain_error]]
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[[kurtosis][3 + 12 * (5[nu] - 22) / (([nu] - 6) * ([nu]-8)) for [nu] >8, else a __domain_error]]
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] [/table]
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[h4 References]
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# Bayesian Data Analysis, Andrew Gelman, John B. Carlin, Hal S. Stern, Donald B. Rubin,
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ISBN-13: 978-1584883883, Chapman & Hall; 2 edition (29 July 2003).
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# Bayesian Computation with R, Jim Albert, ISBN-13: 978-0387922973, Springer; 2nd ed. edition (10 Jun 2009)
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[endsect] [/section:inverse_chi_squared_dist Inverse chi_squared 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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] |