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 is used in Bayesian inference as conjugate prior for the variance of the normal distribution.

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.

If ${\displaystyle X}$ follows a chi-squared distribution with ${\displaystyle \nu }$ degrees of freedom then ${\displaystyle 1/X}$ follows the inverse chi-squared distribution with ${\displaystyle \nu }$ degrees of freedom.

The probability density function of the inverse chi-squared distribution is given by

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

In the above ${\displaystyle x>0}$ and ${\displaystyle \nu }$ is the degrees of freedom parameter. Further, ${\displaystyle \Gamma }$ is the gamma function.

The inverse chi-squared distribution is a special case of the inverse-gamma distribution. with shape parameter ${\displaystyle \alpha ={\frac {\nu }{2}}}$ and scale parameter ${\displaystyle \beta ={\frac {1}{2}}}$.

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

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

• InvChisquare in geoR package for the R Language.