Inverse-chi-squared distribution

Short description: Probability distribution

Inverse-chi-squared

Probability density function
Cumulative distribution function
Parameters	$\nu >0$
Support	$x\in (0,\mathrm{\infty })$
PDF	$\frac{2^{-\nu /2}}{\mathrm{\Gamma }(\nu /2)}{x}^{-\nu /2-1}{e}^{-1/(2x)}$
CDF	$\mathrm{\Gamma }(\frac{\nu }{2},\frac{1}{2x})/\mathrm{\Gamma }\left(\frac{\nu }{2}\right)$
Mean	$\frac{1}{\nu -2}$ for $\nu >2$
Median	$\approx \frac{1}{\nu (1-\frac{2}{9\nu }{)}^3}$
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 $\nu >6$
Kurtosis	$\frac{12(5\nu -22)}{(\nu -6)(\nu -8)}$ for $\nu >8$
Entropy	$\frac{\nu }{2}+\ln \left(\frac{\nu }{2}\mathrm{\Gamma }\left(\frac{\nu }{2}\right)\right)$ $-(1+\frac{\nu }{2})\psi \left(\frac{\nu }{2}\right)$
MGF	$\frac{2}{\mathrm{\Gamma }(\frac{\nu }{2})}{\left(\frac{-t}{2i}\right)}^{\frac{\nu }{4}}{K}_{\frac{\nu }{2}}\left(\sqrt{-2t}\right)$; does not exist as real valued function
CF	$\frac{2}{\mathrm{\Gamma }(\frac{\nu }{2})}{\left(\frac{-it}{2}\right)}^{\frac{\nu }{4}}{K}_{\frac{\nu }{2}}\left(\sqrt{-2it}\right)$

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.

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

$f_1(x;\nu )=\frac{2^{-\nu /2}}{\mathrm{\Gamma }(\nu /2)}{x}^{-\nu /2-1}{e}^{-1/(2x)},$

while the second definition yields the density function

$f_2(x;\nu )=\frac{(\nu /2{)}^{\nu /2}}{\mathrm{\Gamma }(\nu /2)}{x}^{-\nu /2-1}{e}^{-\nu /(2x)}.$

In both cases, $x>0$ and $\nu$ is the degrees of freedom parameter. Further, $\mathrm{\Gamma }$ 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 $X\sim {\chi }^2(\nu )$ and $Y=\frac{1}{X}$, then $Y\sim \text{Inv-}{\chi }^2(\nu )$
• scaled-inverse chi-squared: If $X\sim \text{Scale-inv-}{\chi }^2(\nu ,1/\nu )$, then $X\sim \text{inv-}{\chi }^2(\nu )$
• 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.