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++ +
+#include <boost/math/distributions/inverse_chi_squared.hpp>+
+
+namespace boost{ namespace math{ + +template <class RealType = double, + class Policy = policies::policy<> > +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 + scaled + inverse chi_squared distribution. +
++ A second version has an implicit scale = 1/degrees of freedom and gives + the 1st definition in the 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 The R Project for Statistical Computing + (geoR) with default scale = 1/degrees of freedom. +
++ See +
++ The inverse_chi_squared distribution is used in Bayesian + statistics: the scaled inverse chi-square is conjugate prior + for the normal distribution with known mean, model parameter σ² (variance). +
++ See conjugate + priors including a table of distributions and their priors. +
++ See also Inverse + Gamma Distribution and Chi + Squared Distribution. +
++ The inverse_chi_squared distribution is a psecial case of a inverse_gamma + distribution with nu (degrees_of_freedom) shape (α) and scale (β) where +
++ α= ν /2 and β = ½. +
+![[Note]](../../../../../../../../../doc/src/images/note.png) |
+Note | +
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+ + This distribution does provide the + typedef: + ++ + +typedef inverse_chi_squared_distribution<double> inverse_chi_squared;+ + +
+ If you want a + + +boost::math::inverse_chi_squared_distribution<>+ + +
+ or you can write |
+ For degrees of freedom parameter ν and scale parameter ξ, it is defined + by the probability density function (PDF): +
++ f(x;ν, ξ) = 2 -ν/2 e(-1/2x x (-1-ν/2) / Γ(ν/2) +
++ and Cumulative Density Function (CDF) +
++ F(x;ν, ξ) = Γ( ν /2, νξ/2x) / Γ(ν /2) +
++ The following graphs illustrate how the PDF and CDF of the inverse chi_squared + distribution varies for a few values of parameters ν and ξ: +
+
+ 
+
+ 
+
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 ν degrees of freedom + df, and scale scale with default + value 1df (1ν. +
++ Requires that the degrees of freedom ν parameter is greater than zero, + otherwise calls domain_error. +
+RealType degrees_of_freedom()const; ++
+ Returns the degrees_of_freedom ν parameter of this distribution. +
+RealType scale()const; ++
+ Returns the scale ξ parameter of this distribution. +
++ All the usual non-member + accessor functions that are generic to all distributions are supported: + Cumulative Distribution Function, + Probability Density Function, Quantile, Hazard + Function, Cumulative Hazard Function, + mean, median, + mode, variance, + standard deviation, skewness, + kurtosis, kurtosis_excess, + range and support. +
++ The domain of the random variate is [0,+∞]. +
+ |
+Note | +
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+ Unlike some definitions, this implementation supports a random variate + equal to zero as a special case, returning zero for both pdf and cdf. + |
+ The inverse gamma distribution is implemented in terms of the incomplete + gamma functions like the Inverse + Gamma Distribution 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. +
++ In the following table ν is the degrees of freedom parameter and ξ is the + scale parameter of the distribution, x is the random + variate, p is the probability and q = + 1-p its complement. Parameters α for shape and β for scale are + used for the inverse gamma function: α = ν/2 and β = ν * ξ/2. +
+|
+ + Function + + |
+
+ + Implementation Notes + + |
+
|---|---|
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+ + pdf + + |
+
+ + Using the relation: pdf = gamma_p_derivative(α, + β/ x, β) / x * x + + |
+
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+ + cdf + + |
+
+ + Using the relation: p = gamma_q(α, + β / x) + + |
+
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+ + cdf complement + + |
+
+ + Using the relation: q = gamma_p(α, + β / x) + + |
+
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+ + quantile + + |
+
+ + Using the relation: x = β / gamma_q_inv(α, + p) + + |
+
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+ + quantile from the complement + + |
+
+ + Using the relation: x = α / gamma_p_inv(α, + q) + + |
+
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+ + mode + + |
+
+ + ν * ξ / (ν + 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 + + |
+
+ + 1 / (ν - 1) for ν > 2, else a domain_error + + |
+
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+ + variance + + |
+
+ + 2 ν² ξ² / ((ν -2)² (ν -4)) for ν >4, else a domain_error + + |
+
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+ + skewness + + |
+
+ + 4 √2 √(ν-4) /(ν-6) for ν >6, else a domain_error + + |
+
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+ + kurtosis_excess + + |
+
+ + 12 * (5ν - 22) / (ν - 6) * (ν - 8) for ν >8, else a domain_error + + |
+
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+ + kurtosis + + |
+
+ + 3 + 12 * (5ν - 22) / (ν - 6) * (ν-8) for ν >8, else a domain_error + + |
+
| + | + |
+