# Scale-inverse-chi-square distribution

Parameters Probability density functionNone uploaded yet Cumulative distribution functionNone uploaded yet $\nu >0\,$ $\sigma ^{2}>0\,$ $x\in (0,\infty )$ ${\frac {(\sigma ^{2}\nu /2)^{\nu /2}}{\Gamma (\nu /2)}}~{\frac {\exp \left[{\frac {-\nu \sigma ^{2}}{2x}}\right]}{x^{1+\nu /2}}}$ $\Gamma \left({\frac {\nu }{2}},{\frac {\sigma ^{2}\nu }{2x}}\right)\left/\Gamma \left({\frac {\nu }{2}}\right)\right.$ ${\frac {\nu \sigma ^{2}}{\nu -2}}$ for $\nu >2\,$ ${\frac {\nu \sigma ^{2}}{\nu +2}}$ ${\frac {2\nu ^{2}\sigma ^{4}}{(\nu -2)^{2}(\nu -4)}}$ for $\nu >4\,$ ${\frac {4}{\nu -6}}{\sqrt {2(\nu -4)}}$ for $\nu >6\,$ ${\frac {12(5\nu -22)}{(\nu -6)(\nu -8)}}$ for $\nu >8\,$ ${\frac {\nu }{2}}\!+\!\ln \left({\frac {\sigma ^{2}\nu }{2}}\Gamma \left({\frac {\nu }{2}}\right)\right)$ $\!-\!\left(1\!+\!{\frac {\nu }{2}}\right)\psi \left({\frac {\nu }{2}}\right)$ ${\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-\sigma ^{2}\nu t}{2}}\right)^{\!\!{\frac {\nu }{4}}}\!\!K_{\frac {\nu }{2}}\left({\sqrt {-2\sigma ^{2}\nu t}}\right)$ ${\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-i\sigma ^{2}\nu t}{2}}\right)^{\!\!{\frac {\nu }{4}}}\!\!K_{\frac {\nu }{2}}\left({\sqrt {-2i\sigma ^{2}\nu t}}\right)$ 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

$f(x;\nu ,\sigma ^{2})={\frac {(\sigma ^{2}\nu /2)^{\nu /2}}{\Gamma (\nu /2)}}~{\frac {\exp \left[{\frac {-\nu \sigma ^{2}}{2x}}\right]}{x^{1+\nu /2}}}$ where $\nu$ is the degrees of freedom parameter and $\sigma ^{2}$ 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 $\Gamma (a,x)$ is the incomplete Gamma function, $\Gamma (x)$ is the Gamma function and $Q(a,x)$ is a regularized Gamma function. The characteristic function is

$\varphi (t;\nu ,\sigma ^{2})=$ ${\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-i\sigma ^{2}\nu t}{2}}\right)^{\!\!{\frac {\nu }{4}}}\!\!K_{\frac {\nu }{2}}\left({\sqrt {-2i\sigma ^{2}\nu t}}\right)$ where $K_{\frac {\nu }{2}}(z)$ is the modified Bessel function of the second kind.

## 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 ${\frac {\nu }{2}}$ 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 $\psi (x)$ is the digamma function. An initial estimate can be found by taking the formula for mean and solving it for $\nu .$ Let ${\bar {x}}={\frac {1}{n}}\sum _{i=1}^{N}x_{i}$ be the sample mean. Then an initial estimate for $\nu$ is given by:

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