Scaled inverse chi-squared distribution

Scaled inverse chi-squared
Probability density function
Cumulative distribution function
Parameters	${\displaystyle \nu >0\,}$ ${\displaystyle \tau ^{2}>0\,}$
Support	${\displaystyle x\in (0,\infty )}$
PDF	${\displaystyle {\frac {(\tau ^{2}\nu /2)^{\nu /2}}{\Gamma (\nu /2)}}~{\frac {\exp \left[{\frac {-\nu \tau ^{2}}{2x}}\right]}{x^{1+\nu /2}}}}$
CDF	${\displaystyle \Gamma \left({\frac {\nu }{2}},{\frac {\tau ^{2}\nu }{2x}}\right)\left/\Gamma \left({\frac {\nu }{2}}\right)\right.}$
Mean	${\displaystyle {\frac {\nu \tau ^{2}}{\nu -2}}}$ for ${\displaystyle \nu >2\,}$
Mode	${\displaystyle {\frac {\nu \tau ^{2}}{\nu +2}}}$
Variance	${\displaystyle {\frac {2\nu ^{2}\tau ^{4}}{(\nu -2)^{2}(\nu -4)}}}$for ${\displaystyle \nu >4\,}$
Skewness	${\displaystyle {\frac {4}{\nu -6}}{\sqrt {2(\nu -4)}}}$for ${\displaystyle \nu >6\,}$
Excess kurtosis	${\displaystyle {\frac {12(5\nu -22)}{(\nu -6)(\nu -8)}}}$for ${\displaystyle \nu >8\,}$
Entropy	${\displaystyle {\frac {\nu }{2}}\!+\!\ln \left({\frac {\tau ^{2}\nu }{2}}\Gamma \left({\frac {\nu }{2}}\right)\right)}$ ${\displaystyle \!-\!\left(1\!+\!{\frac {\nu }{2}}\right)\psi \left({\frac {\nu }{2}}\right)}$
MGF	${\displaystyle {\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-\tau ^{2}\nu t}{2}}\right)^{\!\!{\frac {\nu }{4}}}\!\!K_{\frac {\nu }{2}}\left({\sqrt {-2\tau ^{2}\nu t}}\right)}$
CF	${\displaystyle {\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-i\tau ^{2}\nu t}{2}}\right)^{\!\!{\frac {\nu }{4}}}\!\!K_{\frac {\nu }{2}}\left({\sqrt {-2i\tau ^{2}\nu t}}\right)}$

The scaled inverse chi-squared distribution ${\displaystyle \psi \,{\mbox{inv-}}\chi ^{2}(\nu )}$, where ${\displaystyle \psi }$ is the scale parameter, equals the univariate inverse Wishart distribution ${\displaystyle {\mathcal {W}}^{-1}(\psi ,\nu )}$ with degrees of freedom ${\displaystyle \nu }$.

This family of scaled inverse chi-squared distributions is linked to the inverse-chi-squared distribution and to the chi-squared distribution:

If ${\displaystyle X\sim \psi \,{\mbox{inv-}}\chi ^{2}(\nu )}$ then ${\displaystyle X/\psi \sim {\mbox{inv-}}\chi ^{2}(\nu )}$ as well as ${\displaystyle \psi /X\sim \chi ^{2}(\nu )}$ and ${\displaystyle 1/X\sim \psi ^{-1}\chi ^{2}(\nu )}$.

Instead of ${\displaystyle \psi }$, the scaled inverse chi-squared distribution is however most frequently parametrized by the scale parameter ${\displaystyle \tau ^{2}=\psi /\nu }$ and the distribution ${\displaystyle \nu \tau ^{2}\,{\mbox{inv-}}\chi ^{2}(\nu )}$ is denoted by ${\displaystyle {\mbox{Scale-inv-}}\chi ^{2}(\nu ,\tau ^{2})}$.

In terms of ${\displaystyle \tau ^{2}}$ the above relations can be written as follows:

If ${\displaystyle X\sim {\mbox{Scale-inv-}}\chi ^{2}(\nu ,\tau ^{2})}$ then ${\displaystyle {\frac {X}{\nu \tau ^{2}}}\sim {\mbox{inv-}}\chi ^{2}(\nu )}$ as well as ${\displaystyle {\frac {\nu \tau ^{2}}{X}}\sim \chi ^{2}(\nu )}$ and ${\displaystyle 1/X\sim {\frac {1}{\nu \tau ^{2}}}\chi ^{2}(\nu )}$.

This family of scaled inverse chi-squared distributions is a reparametrization of the inverse-gamma distribution.

Specifically, if

${\displaystyle X\sim \psi \,{\mbox{inv-}}\chi ^{2}(\nu )={\mbox{Scale-inv-}}\chi ^{2}(\nu ,\tau ^{2})}$   then   ${\displaystyle X\sim {\textrm {Inv-Gamma}}\left({\frac {\nu }{2}},{\frac {\psi }{2}}\right)={\textrm {Inv-Gamma}}\left({\frac {\nu }{2}},{\frac {\nu \tau ^{2}}{2}}\right)}$

Either form may be used to represent the maximum entropy distribution for a fixed first inverse moment ${\displaystyle (E(1/X))}$ and first logarithmic moment ${\displaystyle (E(\ln(X))}$.

The scaled inverse chi-squared distribution also has a particular use in Bayesian statistics. Specifically, the scaled inverse chi-squared distribution can be used as a conjugate prior for the variance parameter of a normal distribution. The same prior in alternative parametrization is given by the inverse-gamma distribution.