1[section:inverse_chi_squared_dist Inverse Chi Squared Distribution] 2 3``#include <boost/math/distributions/inverse_chi_squared.hpp>`` 4 5 namespace boost{ namespace math{ 6 7 template <class RealType = double, 8 class ``__Policy`` = ``__policy_class`` > 9 class inverse_chi_squared_distribution 10 { 11 public: 12 typedef RealType value_type; 13 typedef Policy policy_type; 14 15 inverse_chi_squared_distribution(RealType df = 1); // Not explicitly scaled, default 1/df. 16 inverse_chi_squared_distribution(RealType df, RealType scale = 1/df); // Scaled. 17 18 RealType degrees_of_freedom()const; // Default 1. 19 RealType scale()const; // Optional scale [xi] (variance), default 1/degrees_of_freedom. 20 }; 21 22 }} // namespace boost // namespace math 23 24The inverse chi squared distribution is a continuous probability distribution 25of the *reciprocal* of a variable distributed according to the chi squared distribution. 26 27The sources below give confusingly different formulae 28using different symbols for the distribution pdf, 29but they are all the same, or related by a change of variable, or choice of scale. 30 31Two constructors are available to implement both the scaled and (implicitly) unscaled versions. 32 33The main version has an explicit scale parameter which implements the 34[@http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution scaled inverse chi_squared distribution]. 35 36A second version has an implicit scale = 1/degrees of freedom and gives the 1st definition in the 37[@http://en.wikipedia.org/wiki/Inverse-chi-square_distribution Wikipedia inverse chi_squared distribution]. 38The 2nd Wikipedia inverse chi_squared distribution definition can be implemented 39by explicitly specifying a scale = 1. 40 41Both definitions are also available in __Mathematica and in __R (geoR) with default scale = 1/degrees of freedom. 42 43See 44 45* Inverse chi_squared distribution [@http://en.wikipedia.org/wiki/Inverse-chi-square_distribution] 46* Scaled inverse chi_squared distribution[@http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution] 47* R inverse chi_squared distribution functions [@http://hosho.ees.hokudai.ac.jp/~kubo/Rdoc/library/geoR/html/InvChisquare.html R ] 48* 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.] 49* Inverse chi_squared distribution reference [@http://reference.wolfram.com/mathematica/ref/InverseChiSquareDistribution.html Weisstein, Eric W. "Inverse Chi-Squared Distribution reference." From Wolfram Mathematica.] 50 51The inverse_chi_squared distribution is used in 52[@http://en.wikipedia.org/wiki/Bayesian_statistics Bayesian statistics]: 53the scaled inverse chi-square is conjugate prior for the normal distribution 54with known mean, model parameter [sigma][pow2] (variance). 55 56See [@http://en.wikipedia.org/wiki/Conjugate_prior conjugate priors including a table of distributions and their priors.] 57 58See also __inverse_gamma_distrib and __chi_squared_distrib. 59 60The inverse_chi_squared distribution is a special case of a inverse_gamma distribution 61with [nu] (degrees_of_freedom) shape ([alpha]) and scale ([beta]) where 62 63[expression [alpha]= [nu] /2 and [beta] = [frac12]] 64 65[note This distribution *does* provide the typedef: 66 67``typedef inverse_chi_squared_distribution<double> inverse_chi_squared;`` 68 69If you want a `double` precision inverse_chi_squared distribution you can use 70 71``boost::math::inverse_chi_squared_distribution<>`` 72 73or you can write `inverse_chi_squared my_invchisqr(2, 3);`] 74 75For degrees of freedom parameter [nu], 76the (*unscaled*) inverse chi_squared distribution is defined by the probability density function (PDF): 77 78[expression f(x;[nu]) = 2[super -[nu]/2] x[super -[nu]/2-1] e[super -1/2x] / [Gamma]([nu]/2)] 79 80and Cumulative Density Function (CDF) 81 82[expression F(x;[nu]) = [Gamma]([nu]/2, 1/2x) / [Gamma]([nu]/2)] 83 84For degrees of freedom parameter [nu] and scale parameter [xi], 85the *scaled* inverse chi_squared distribution is defined by the probability density function (PDF): 86 87[expression f(x;[nu], [xi]) = ([xi][nu]/2)[super [nu]/2] e[super -[nu][xi]/2x] x[super -1-[nu]/2] / [Gamma]([nu]/2)] 88 89and Cumulative Density Function (CDF) 90 91[expression F(x;[nu], [xi]) = [Gamma]([nu]/2, [nu][xi]/2x) / [Gamma]([nu]/2)] 92 93The following graphs illustrate how the PDF and CDF of the inverse chi_squared distribution 94varies for a few values of parameters [nu] and [xi]: 95 96[graph inverse_chi_squared_pdf] [/.png or .svg] 97 98[graph inverse_chi_squared_cdf] 99 100[h4 Member Functions] 101 102 inverse_chi_squared_distribution(RealType df = 1); // Implicitly scaled 1/df. 103 inverse_chi_squared_distribution(RealType df = 1, RealType scale); // Explicitly scaled. 104 105Constructs an inverse chi_squared distribution with [nu] degrees of freedom ['df], 106and scale ['scale] with default value 1\/df. 107 108Requires that the degrees of freedom [nu] parameter is greater than zero, otherwise calls 109__domain_error. 110 111 RealType degrees_of_freedom()const; 112 113Returns the degrees_of_freedom [nu] parameter of this distribution. 114 115 RealType scale()const; 116 117Returns the scale [xi] parameter of this distribution. 118 119[h4 Non-member Accessors] 120 121All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all 122distributions are supported: __usual_accessors. 123 124The domain of the random variate is \[0,+[infin]\]. 125[note Unlike some definitions, this implementation supports a random variate 126equal to zero as a special case, returning zero for both pdf and cdf.] 127 128[h4 Accuracy] 129 130The inverse gamma distribution is implemented in terms of the 131incomplete gamma functions like the __inverse_gamma_distrib that use 132__gamma_p and __gamma_q and their inverses __gamma_p_inv and __gamma_q_inv: 133refer to the accuracy data for those functions for more information. 134But in general, gamma (and thus inverse gamma) results are often accurate to a few epsilon, 135>14 decimal digits accuracy for 64-bit double. 136unless iteration is involved, as for the estimation of degrees of freedom. 137 138[h4 Implementation] 139 140In the following table [nu] is the degrees of freedom parameter and 141[xi] is the scale parameter of the distribution, 142/x/ is the random variate, /p/ is the probability and /q = 1-p/ its complement. 143Parameters [alpha] for shape and [beta] for scale 144are used for the inverse gamma function: [alpha] = [nu]/2 and [beta] = [nu] * [xi]/2. 145 146[table 147[[Function][Implementation Notes]] 148[[pdf][Using the relation: pdf = __gamma_p_derivative([alpha], [beta]/ x, [beta]) / x * x ]] 149[[cdf][Using the relation: p = __gamma_q([alpha], [beta] / x) ]] 150[[cdf complement][Using the relation: q = __gamma_p([alpha], [beta] / x) ]] 151[[quantile][Using the relation: x = [beta]/ __gamma_q_inv([alpha], p) ]] 152[[quantile from the complement][Using the relation: x = [alpha]/ __gamma_p_inv([alpha], q) ]] 153[[mode][[nu] * [xi] / ([nu] + 2) ]] 154[[median][no closed form analytic equation is known, but is evaluated as quantile(0.5)]] 155[[mean][[nu][xi] / ([nu] - 2) for [nu] > 2, else a __domain_error]] 156[[variance][2 [nu][pow2] [xi][pow2] / (([nu] -2)[pow2] ([nu] -4)) for [nu] >4, else a __domain_error]] 157[[skewness][4 [sqrt]2 [sqrt]([nu]-4) /([nu]-6) for [nu] >6, else a __domain_error ]] 158[[kurtosis_excess][12 * (5[nu] - 22) / (([nu] - 6) * ([nu] - 8)) for [nu] >8, else a __domain_error]] 159[[kurtosis][3 + 12 * (5[nu] - 22) / (([nu] - 6) * ([nu]-8)) for [nu] >8, else a __domain_error]] 160] [/table] 161 162[h4 References] 163 164# Bayesian Data Analysis, Andrew Gelman, John B. Carlin, Hal S. Stern, Donald B. Rubin, 165ISBN-13: 978-1584883883, Chapman & Hall; 2 edition (29 July 2003). 166 167# Bayesian Computation with R, Jim Albert, ISBN-13: 978-0387922973, Springer; 2nd ed. edition (10 Jun 2009) 168 169[endsect] [/section:inverse_chi_squared_dist Inverse chi_squared Distribution] 170 171[/ 172 Copyright 2010 John Maddock and Paul A. Bristow. 173 Distributed under the Boost Software License, Version 1.0. 174 (See accompanying file LICENSE_1_0.txt or copy at 175 http://www.boost.org/LICENSE_1_0.txt). 176]