| [section:inverse_chi_squared_dist Inverse Chi Squared Distribution] |
| |
| ``#include <boost/math/distributions/inverse_chi_squared.hpp>`` |
| |
| namespace boost{ namespace math{ |
| |
| template <class RealType = double, |
| class ``__Policy`` = ``__policy_class`` > |
| class inverse_chi_squared_distribution |
| { |
| public: |
| typedef RealType value_type; |
| typedef Policy policy_type; |
| |
| inverse_chi_squared_distribution(RealType df = 1); // Not explicitly scaled, default 1/df. |
| inverse_chi_squared_distribution(RealType df, RealType scale = 1/df); // Scaled. |
| |
| RealType degrees_of_freedom()const; // Default 1. |
| RealType scale()const; // Optional scale [xi] (variance), default 1/degrees_of_freedom. |
| }; |
| |
| }} // namespace boost // namespace math |
| |
| The inverse chi squared distribution is a continuous probability distribution |
| of the *reciprocal* of a variable distributed according to the chi squared distribution. |
| |
| The sources below give confusingly different formulae |
| using different symbols for the distribution pdf, |
| but they are all the same, or related by a change of variable, or choice of scale. |
| |
| Two constructors are available to implement both the scaled and (implicitly) unscaled versions. |
| |
| The main version has an explicit scale parameter which implements the |
| [@http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution scaled inverse chi_squared distribution]. |
| |
| A second version has an implicit scale = 1/degrees of freedom and gives the 1st definition in the |
| [@http://en.wikipedia.org/wiki/Inverse-chi-square_distribution Wikipedia inverse chi_squared distribution]. |
| The 2nd Wikipedia inverse chi_squared distribution definition can be implemented |
| by explicitly specifying a scale = 1. |
| |
| Both definitions are also available in Wolfram Mathematica and in __R (geoR) with default scale = 1/degrees of freedom. |
| |
| See |
| |
| * Inverse chi_squared distribution [@http://en.wikipedia.org/wiki/Inverse-chi-square_distribution] |
| * Scaled inverse chi_squared distribution[@http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution] |
| * R inverse chi_squared distribution functions [@http://hosho.ees.hokudai.ac.jp/~kubo/Rdoc/library/geoR/html/InvChisquare.html R ] |
| * 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.] |
| * Inverse chi_squared distribution reference [@http://reference.wolfram.com/mathematica/ref/InverseChiSquareDistribution.html Weisstein, Eric W. "Inverse Chi-Squared Distribution reference." From Wolfram Mathematica.] |
| |
| The inverse_chi_squared distribution is used in |
| [@http://en.wikipedia.org/wiki/Bayesian_statistics Bayesian statistics]: |
| the scaled inverse chi-square is conjugate prior for the normal distribution |
| with known mean, model parameter [sigma][pow2] (variance). |
| |
| See [@http://en.wikipedia.org/wiki/Conjugate_prior conjugate priors including a table of distributions and their priors.] |
| |
| See also __inverse_gamma_distrib and __chi_squared_distrib. |
| |
| The inverse_chi_squared distribution is a special case of a inverse_gamma distribution |
| with [nu] (degrees_of_freedom) shape ([alpha]) and scale ([beta]) where |
| |
| __spaces [alpha]= [nu] /2 and [beta] = [frac12]. |
| |
| [note This distribution *does* provide the typedef: |
| |
| ``typedef inverse_chi_squared_distribution<double> inverse_chi_squared;`` |
| |
| If you want a `double` precision inverse_chi_squared distribution you can use |
| |
| ``boost::math::inverse_chi_squared_distribution<>`` |
| |
| or you can write `inverse_chi_squared my_invchisqr(2, 3);`] |
| |
| For degrees of freedom parameter [nu], |
| the (*unscaled*) inverse chi_squared distribution is defined by the probability density function (PDF): |
| |
| __spaces f(x;[nu]) = 2[super -[nu]/2] x[super -[nu]/2-1] e[super -1/2x] / [Gamma]([nu]/2) |
| |
| and Cumulative Density Function (CDF) |
| |
| __spaces F(x;[nu]) = [Gamma]([nu]/2, 1/2x) / [Gamma]([nu]/2) |
| |
| For degrees of freedom parameter [nu] and scale parameter [xi], |
| the *scaled* inverse chi_squared distribution is defined by the probability density function (PDF): |
| |
| __spaces f(x;[nu], [xi]) = ([xi][nu]/2)[super [nu]/2] e[super -[nu][xi]/2x] x[super -1-[nu]/2] / [Gamma]([nu]/2) |
| |
| and Cumulative Density Function (CDF) |
| |
| __spaces F(x;[nu], [xi]) = [Gamma]([nu]/2, [nu][xi]/2x) / [Gamma]([nu]/2) |
| |
| The following graphs illustrate how the PDF and CDF of the inverse chi_squared distribution |
| varies for a few values of parameters [nu] and [xi]: |
| |
| [graph inverse_chi_squared_pdf] [/.png or .svg] |
| |
| [graph inverse_chi_squared_cdf] |
| |
| [h4 Member Functions] |
| |
| inverse_chi_squared_distribution(RealType df = 1); // Implicitly scaled 1/df. |
| inverse_chi_squared_distribution(RealType df = 1, RealType scale); // Explicitly scaled. |
| |
| Constructs an inverse chi_squared distribution with [nu] degrees of freedom ['df], |
| and scale ['scale] with default value 1\/df. |
| |
| Requires that the degrees of freedom [nu] parameter is greater than zero, otherwise calls |
| __domain_error. |
| |
| RealType degrees_of_freedom()const; |
| |
| Returns the degrees_of_freedom [nu] parameter of this distribution. |
| |
| RealType scale()const; |
| |
| Returns the scale [xi] parameter of this distribution. |
| |
| [h4 Non-member Accessors] |
| |
| All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] that are generic to all |
| distributions are supported: __usual_accessors. |
| |
| The domain of the random variate is \[0,+[infin]\]. |
| [note Unlike some definitions, this implementation supports a random variate |
| equal to zero as a special case, returning zero for both pdf and cdf.] |
| |
| [h4 Accuracy] |
| |
| The inverse gamma distribution is implemented in terms of the |
| incomplete gamma functions like the __inverse_gamma_distrib that use |
| __gamma_p and __gamma_q and their inverses __gamma_p_inv and __gamma_q_inv: |
| refer to the accuracy data for those functions for more information. |
| But in general, gamma (and thus inverse gamma) results are often accurate to a few epsilon, |
| >14 decimal digits accuracy for 64-bit double. |
| unless iteration is involved, as for the estimation of degrees of freedom. |
| |
| [h4 Implementation] |
| |
| In the following table [nu] is the degrees of freedom parameter and |
| [xi] is the scale parameter of the distribution, |
| /x/ is the random variate, /p/ is the probability and /q = 1-p/ its complement. |
| Parameters [alpha] for shape and [beta] for scale |
| are used for the inverse gamma function: [alpha] = [nu]/2 and [beta] = [nu] * [xi]/2. |
| |
| [table |
| [[Function][Implementation Notes]] |
| [[pdf][Using the relation: pdf = __gamma_p_derivative([alpha], [beta]/ x, [beta]) / x * x ]] |
| [[cdf][Using the relation: p = __gamma_q([alpha], [beta] / x) ]] |
| [[cdf complement][Using the relation: q = __gamma_p([alpha], [beta] / x) ]] |
| [[quantile][Using the relation: x = [beta][space]/ __gamma_q_inv([alpha], p) ]] |
| [[quantile from the complement][Using the relation: x = [alpha][space]/ __gamma_p_inv([alpha], q) ]] |
| [[mode][[nu] * [xi] / ([nu] + 2) ]] |
| [[median][no closed form analytic equation is known, but is evaluated as quantile(0.5)]] |
| [[mean][[nu][xi] / ([nu] - 2) for [nu] > 2, else a __domain_error]] |
| [[variance][2 [nu][pow2] [xi][pow2] / (([nu] -2)[pow2] ([nu] -4)) for [nu] >4, else a __domain_error]] |
| [[skewness][4 [sqrt]2 [sqrt]([nu]-4) /([nu]-6) for [nu] >6, else a __domain_error ]] |
| [[kurtosis_excess][12 * (5[nu] - 22) / (([nu] - 6) * ([nu] - 8)) for [nu] >8, else a __domain_error]] |
| [[kurtosis][3 + 12 * (5[nu] - 22) / (([nu] - 6) * ([nu]-8)) for [nu] >8, else a __domain_error]] |
| ] [/table] |
| |
| [h4 References] |
| |
| # Bayesian Data Analysis, Andrew Gelman, John B. Carlin, Hal S. Stern, Donald B. Rubin, |
| ISBN-13: 978-1584883883, Chapman & Hall; 2 edition (29 July 2003). |
| |
| # Bayesian Computation with R, Jim Albert, ISBN-13: 978-0387922973, Springer; 2nd ed. edition (10 Jun 2009) |
| |
| [endsect] [/section:inverse_chi_squared_dist Inverse chi_squared Distribution] |
| |
| [/ |
| Copyright 2010 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). |
| ] |