Inverse-chi-squared distribution

In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distribution) is a continuous probability distribution of a positive-valued random variable. It is closely related to the chi-squared distribution. It arises in Bayesian inference, where it can be used as the prior and posterior distribution for an unknown variance of the normal distribution.

Inverse-chi-squared

Probability density function
Cumulative distribution function
Parameters	$\nu >0\!$
Support	$x\in (0,\infty )\!$
PDF	${\frac {2^{-\nu /2}}{\Gamma (\nu /2)}}\,x^{-\nu /2-1}e^{-1/(2x)}\!$
CDF	$\Gamma \!\left({\frac {\nu }{2}},{\frac {1}{2x}}\right){\bigg /}\,\Gamma \!\left({\frac {\nu }{2}}\right)\!$
Mean	${\frac {1}{\nu -2}}\!$ for $\nu >2\!$
Median	$\approx {\dfrac {1}{\nu {\bigg (}1-{\dfrac {2}{9\nu }}{\bigg )}^{3}}}$
Mode	${\frac {1}{\nu +2}}\!$
Variance	${\frac {2}{(\nu -2)^{2}(\nu -4)}}\!$ 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	${\frac {\nu }{2}}\!+\!\ln \!\left({\frac {\nu }{2}}\Gamma \!\left({\frac {\nu }{2}}\right)\right)$ $\!-\!\left(1\!+\!{\frac {\nu }{2}}\right)\psi \!\left({\frac {\nu }{2}}\right)$
MGF	${\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-t}{2i}}\right)^{\!\!{\frac {\nu }{4}}}K_{\frac {\nu }{2}}\!\left({\sqrt {-2t}}\right)$; does not exist as real valued function
CF	${\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-it}{2}}\right)^{\!\!{\frac {\nu }{4}}}K_{\frac {\nu }{2}}\!\left({\sqrt {-2it}}\right)$

Definition

The inverse-chi-squared distribution (or inverted-chi-square distribution ) is the probability distribution of a random variable whose multiplicative inverse (reciprocal) has a chi-squared distribution. It is also often defined as the distribution of a random variable whose reciprocal divided by its degrees of freedom is a chi-squared distribution. That is, if $X$ has the chi-squared distribution with $\nu$ degrees of freedom, then according to the first definition, $1/X$ has the inverse-chi-squared distribution with $\nu$ degrees of freedom; while according to the second definition, $\nu /X$ has the inverse-chi-squared distribution with $\nu$ degrees of freedom. Information associated with the first definition is depicted on the right side of the page.

The first definition yields a probability density function given by

$f_{1}(x;\nu )={\frac {2^{-\nu /2}}{\Gamma (\nu /2)}}\,x^{-\nu /2-1}e^{-1/(2x)},$

while the second definition yields the density function

$f_{2}(x;\nu )={\frac {(\nu /2)^{\nu /2}}{\Gamma (\nu /2)}}x^{-\nu /2-1}e^{-\nu /(2x)}.$

In both cases, $x>0$ and $\nu$ is the degrees of freedom parameter. Further, $\Gamma$ is the gamma function. Both definitions are special cases of the scaled-inverse-chi-squared distribution. For the first definition the variance of the distribution is $\sigma ^{2}=1/\nu ,$ while for the second definition $\sigma ^{2}=1$.

Related distributions

• chi-squared: If $X\thicksim \chi ^{2}(\nu )$ and $Y={\frac {1}{X}}$, then $Y\thicksim {\text{Inv-}}\chi ^{2}(\nu )$
• scaled-inverse chi-squared: If $X\thicksim {\text{Scale-inv-}}\chi ^{2}(\nu ,1/\nu )\,$, then $X\thicksim {\text{inv-}}\chi ^{2}(\nu )$
• Inverse gamma with $\alpha ={\frac {\nu }{2}}$ and $\beta ={\frac {1}{2}}$
• Inverse chi-squared distribution is a special case of type 5 Pearson distribution

See also

• Scaled-inverse-chi-squared distribution
• Inverse-Wishart distribution

External links

• InvChisquare in geoR package for the R Language.