# Scale-inverse-chi-square distribution

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

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 ${\displaystyle x>0}$ is

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

where ${\displaystyle \nu }$ is the degrees of freedom parameter and ${\displaystyle \sigma ^{2}}$ is the scale parameter. The cumulative distribution function is

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

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

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

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

## Parameter estimation

The maximum likelihood estimate of ${\displaystyle \sigma ^{2}}$ is

${\displaystyle \sigma ^{2}=n/\sum _{i=1}^{N}{\frac {1}{x_{i}}}.}$

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

${\displaystyle \ln({\frac {\nu }{2}})+\psi ({\frac {\nu }{2}})=\sum _{i=1}^{N}\ln(x_{i})-n\ln(\sigma ^{2})}$

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

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

## Related distributions

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