Inverse-chi-square distribution

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Inverse-chi-square
Probability density function
Image:Inverse chi squared density.png
Cumulative distribution function
Image:Inverse chi squared distribution.png
Parameters \nu > 0\!
Support x \in (0, \infty)\!
Probability density function (pdf) \frac{2^{-\nu/2}}{\Gamma(\nu/2)}\,x^{-\nu/2-1} e^{-1/(2 x)}\!
Cumulative distribution function (cdf) \Gamma\!\left(\frac{\nu}{2},\frac{1}{2x}\right) \bigg/\, \Gamma\!\left(\frac{\nu}{2}\right)\!
Mean \frac{1}{\nu-2}\! for \nu >2\!
Median
Mode \frac{1}{\nu+2}\!
Variance \frac{2}{(\nu-2)^2 (\nu-4)}\! for \nu >4\!
Skewness \frac{4}{\nu-6}\sqrt{2(\nu-4)}\! for \nu >6\!
Excess kurtosis \frac{12(5\nu-22)}{(\nu-6)(\nu-8)}\! for \nu >8\!
Entropy \frac{\nu}{2} \!+\!\ln\!\left(\frac{1}{2}\Gamma\!\left(\frac{\nu}{2}\right)\right)

\!-\!\left(1\!+\!\frac{\nu}{2}\right)\psi\!\left(\frac{\nu}{2}\right)
Moment-generating function (mgf) \frac{2}{\Gamma(\frac{\nu}{2})} \left(\frac{-t}{2i}\right)^{\!\!\frac{\nu}{4}} K_{\frac{\nu}{2}}\!\left(\sqrt{-2t}\right)
Characteristic function \frac{2}{\Gamma(\frac{\nu}{2})} \left(\frac{-it}{2}\right)^{\!\!\frac{\nu}{4}} K_{\frac{\nu}{2}}\!\left(\sqrt{-2it}\right)

In probability and statistics, the inverse-chi-square distribution is the probability distribution of a random variable whose inverse has a chi-square distribution. It is also often defined as the distribution of a random variable whose inverse divided by its degrees of freedom is a chi-square distribution. That is, if X has the chi-square distribution with ν degrees of freedom, then according to the first definition, 1 / X has the inverse-chi-square distribution with ν degrees of freedom; while according to the second definition, ν / X has the inverse-chi-square distribution with ν degrees of freedom.

This distribution arises in Bayesian statistics.

It is a continuous distribution with a probability density function. The first definition yields a density function

f(x; \nu) = \frac{2^{-\nu/2}}{\Gamma(\nu/2)}\,x^{-\nu/2-1} e^{-1/(2 x)}

The second definition yields a density function

f(x; \nu) = \frac{(\nu/2)^{\nu/2}}{\Gamma(\nu/2)} x^{-\nu/2-1} e^{-\nu/(2 x)}

In both cases, x > 0 and ν is the degrees of freedom parameter. This article will deal with the first definition only. Both definitions are special cases of the scale-inverse-chi-square distribution. For the first definition σ2 = 1 / ν and for the second definition σ2 = 1.

[edit] Related distributions

[edit] See also

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Univariate Multivariate
Discrete: Benford • Bernoulli • binomial • Boltzmann • categorical • compound Poisson • degenerate • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot Ewens • multinomial • multivariate Polya
Continuous: Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Erlang • exponential • exponential power • F • fading • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square (scaled inverse chi-square)• inverse Gaussian • inverse gamma (scaled inverse gamma) • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • normal (Gaussian) • normal inverse Gaussian • Pareto • Pearson • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • shifted Gompertz • Student's t • triangular • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda Dirichlet • inverse-Wishart • Kent • matrix normal • multivariate normal • multivariate Student • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous: Cantor • conditional • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • phase-type • posterior • prior • quasi • sampling • singular
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