Scale-inverse-chi-square distribution

Probability distribution
name =Scale-inverse-chi-square
type =density
pdf_

cdf_

parameters =$u\; >\; 0,$
$sigma^2\; >\; 0,$
support =$x\; in\; (0,\; infty)$
pdf =$frac\{(sigma^2\; u/2)^\{\; u/2${Gamma( u/2)}~frac{expleft [ frac{- u sigma^2}{2 x} ight] }{x^{1+ u/2
cdf =$Gammaleft(frac\{\; u\}\{2\},frac\{sigma^2\; u\}\{2x\}\; ight)left/Gammaleft(frac\{\; u\}\{2\}\; ight)\; ight.$
mean =$frac\{\; u\; sigma^2\}\{\; u-2\}$ for $u\; >2,$
median =
mode =$frac\{\; u\; sigma^2\}\{\; u+2\}$
variance =$frac\{2\; u^2\; sigma^4\}\{(\; u-2)^2\; (\; u-4)\}$for $u\; >4,$
skewness =$frac\{4\}\{\; u-6\}sqrt\{2(\; u-4)\}$for $u\; >6,$
kurtosis =$frac\{12(5\; u-22)\}\{(\; u-6)(\; u-8)\}$for $u\; >8,$
entropy =$frac\{\; u\}\{2\}!+!lnleft(frac\{sigma^2\; u\}\{2\}Gammaleft(frac\{\; u\}\{2\}\; ight)\; ight)$$!-!left(1!+!frac\{\; u\}\{2\}\; ight)psileft(frac\{\; u\}\{2\}\; ight)$
mgf =$frac\{2\}\{Gamma(frac\{\; u\}\{2\})\}left(frac\{-sigma^2\; u\; t\}\{2\}\; ight)^\{!!frac\{\; u\}\{4!!K\_\{frac\{\; u\}\{2left(sqrt\{-2sigma^2\; u\; t\}\; ight)$
char =$frac\{2\}\{Gamma(frac\{\; u\}\{2\})\}left(frac\{-isigma^2\; u\; t\}\{2\}\; ight)^\{!!frac\{\; u\}\{4!!K\_\{frac\{\; u\}\{2left(sqrt\{-2isigma^2\; u\; t\}\; ight)$
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

:$f(x;\; u,\; sigma^2)=frac\{(sigma^2\; u/2)^\{\; u/2\{Gamma(\; u/2)\}~frac\{expleft\; [\; frac\{-\; u\; sigma^2\}\{2\; x\}\; ight]\; \}\{x^\{1+\; u/2$

where $u$ is the degrees of freedom parameter and $sigma^2$ is the scale parameter. The cumulative distribution function is

:$F(x;\; u,\; sigma^2)=Gammaleft(frac\{\; u\}\{2\},frac\{sigma^2\; u\}\{2x\}\; ight)left/Gammaleft(frac\{\; u\}\{2\}\; ight)\; ight.$:$=Qleft(frac\{\; u\}\{2\},frac\{sigma^2\; u\}\{2x\}\; ight)$

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

:$varphi(t;\; u,sigma^2)=$:$frac\{2\}\{Gamma(frac\{\; u\}\{2\})\}left(frac\{-isigma^2\; u\; t\}\{2\}\; ight)^\{!!frac\{\; u\}\{4!!K\_\{frac\{\; u\}\{2left(sqrt\{-2isigma^2\; u\; t\}\; ight)$

where $K\_\{frac\{\; u\}\{2(z)$ is the modified Bessel function of the second kind.

Parameter estimation

The maximum likelihood estimate of $sigma^2$ is

:$sigma^2\; =\; n/sum\_\{i=1\}^N\; frac\{1\}\{x\_i\}.$

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

:$ln(frac\{\; u\}\{2\})\; +\; psi(frac\{\; u\}\{2\})\; =\; sum\_\{i=1\}^N\; ln(x\_i)\; -\; n\; ln(sigma^2)$

where $psi(x)$ is the digamma function. An initial estimate can be found by taking the formula for mean and solving it for $u.$ Let be the sample mean. Then an initial estimate for $u$ is given by:

:

Related distributions

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

ee also

*Inverse chi-square distribution
*Chi-square distribution
*Bayesian probability