Scale-inverse-chi-square distribution

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Scale-inverse-chi-square
Probability density function None uploaded yet
Cumulative distribution function None uploaded yet
Parameters	$\nu > 0\,$ $\sigma^2 > 0\,$
Support	$x \in (0, \infty)$
Probability density function (pdf)	$\frac{(\sigma^2\nu/2)^{\nu/2}}{\Gamma(\nu/2)}~ \frac{\exp\left[ \frac{-\nu \sigma^2}{2 x}\right]}{x^{1+\nu/2}}$
Cumulative distribution function (cdf)	$\Gamma\left(\frac{\nu}{2},\frac{\sigma^2\nu}{2x}\right) \left/\Gamma\left(\frac{\nu}{2}\right)\right.$
Mean	$\frac{\nu \sigma^2}{\nu-2}$ for $\nu >2\,$
Median
Mode	$\frac{\nu \sigma^2}{\nu+2}$
Variance	$\frac{2 \nu^2 \sigma^4}{(\nu-2)^2 (\nu-4)}$ 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	$\frac{\nu}{2} \!+\!\ln\left(\frac{\sigma^2\nu}{2}\Gamma\left(\frac{\nu}{2}\right)\right)$ $\!-\!\left(1\!+\!\frac{\nu}{2}\right)\psi\left(\frac{\nu}{2}\right)$
Moment-generating function (mgf)	$\frac{2}{\Gamma(\frac{\nu}{2})}\left(\frac{-\sigma^2\nu t}{2}\right)^{\!\!\frac{\nu}{4}}\!\!K_{\frac{\nu}{2}}\left(\sqrt{-2\sigma^2\nu t}\right)$
Characteristic function	$\frac{2}{\Gamma(\frac{\nu}{2})}\left(\frac{-i\sigma^2\nu t}{2}\right)^{\!\!\frac{\nu}{4}}\!\!K_{\frac{\nu}{2}}\left(\sqrt{-2i\sigma^2\nu t}\right)$

The scaled inverse chi-square distribution arises in Bayesian statistics. It is a more general distribution than the inverse-chi-square distribution. Its probability density function over the domain $x > 0$ is

$f(x; \nu, \sigma^2)= \frac{(\sigma^2\nu/2)^{\nu/2}}{\Gamma(\nu/2)}~ \frac{\exp\left[ \frac{-\nu \sigma^2}{2 x}\right]}{x^{1+\nu/2}}$

where $ν$ is the degrees of freedom parameter and $σ 2$ is the scale parameter. The cumulative distribution function is

$F(x; \nu, \sigma^2)= \Gamma\left(\frac{\nu}{2},\frac{\sigma^2\nu}{2x}\right) \left/\Gamma\left(\frac{\nu}{2}\right)\right.$

$=Q\left(\frac{\nu}{2},\frac{\sigma^2\nu}{2x}\right)$

where $Γ(a, x)$ is the incomplete Gamma function, $Γ(x)$ is the Gamma function and $Q (a, x)$ is a regularized Gamma function. The characteristic function is

$\varphi(t;\nu,\sigma^2)=$

$\frac{2}{\Gamma(\frac{\nu}{2})}\left(\frac{-i\sigma^2\nu t}{2}\right)^{\!\!\frac{\nu}{4}}\!\!K_{\frac{\nu}{2}}\left(\sqrt{-2i\sigma^2\nu t}\right)$

where $K_{\frac{\nu}{2}}(z)$ is the modified Bessel function of the second kind.

[edit] Parameter estimation

The maximum likelihood estimate of $σ 2$ is

$\sigma^2 = n/\sum_{i=1}^N \frac{1}{x_i}.$

The maximum likelihood estimate of $\frac{\nu}{2}$ can be found using Newton's method on:

$\ln(\frac{\nu}{2}) + \psi(\frac{\nu}{2}) = \sum_{i=1}^N \ln(x_i) - n \ln(\sigma^2)$

where $ψ(x)$ is the digamma function. An initial estimate can be found by taking the formula for mean and solving it for $ν.$ Let $\bar{x} = \frac{1}{n}\sum_{i=1}^N x_i$ be the sample mean. Then an estimate for $ν$ is:

$\frac{\nu}{2} = \frac{\bar{x}}{\bar{x} - \sigma^2}.$

[edit] Related distributions

Relation to chi-square distribution: If $X \sim \chi^2(\nu)$ and $Y = \frac{\sigma^2 \nu}{X}$ then $Y \sim \mbox{Scale-inv-}\chi^2(\nu, \sigma^2)$
Relation to the inverse gamma distribution: If $X \sim \textrm{Inv-Gamma}\left(\frac{\nu}{2}, \frac{\nu \sigma^2}{2}\right)$ then $X \sim \mbox{Scale-inv-}\chi^2(\nu, \sigma^2)$ .
The scale-inverse-chi-square distribution is a conjugate prior for the variance parameter of a normal distribution.

[edit] See also

	Probability distributions [ view • talk • edit ]
	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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Category: Continuous distributions

Scale-inverse-chi-square distribution

From Wikipedia, the free encyclopedia

[edit] Parameter estimation

[edit] Related distributions

[edit] See also

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