Inverse-chi-squared distribution

Inverse-chi-squared

Probability density function
Cumulative distribution function
Parameters	$\nu > 0\!$
Support	$x \in (0, \infty)\!$
PDF	$\frac{2^{-\nu/2}}{\Gamma(\nu/2)}\,x^{-\nu/2-1} e^{-1/(2 x)}\!$
CDF	$\Gamma\!\left(\frac{\nu}{2},\frac{1}{2x}\right) \bigg/\, \Gamma\!\left(\frac{\nu}{2}\right)\!$
Mean	$\frac{1}{\nu-2}\!$ for $\nu >2\!$
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 $\nu >6\!$
Ex. kurtosis	$\frac{12(5\nu-22)}{(\nu-6)(\nu-8)}\!$ for $\nu >8\!$
Entropy	$\frac{\nu}{2} \!+\!\ln\!\left(\frac{1}{2}\Gamma\!\left(\frac{\nu}{2}\right)\right)$ $\!-\!\left(1\!+\!\frac{\nu}{2}\right)\psi\!\left(\frac{\nu}{2}\right)$
MGF	$\frac{2}{\Gamma(\frac{\nu}{2})} \left(\frac{-t}{2i}\right)^{\!\!\frac{\nu}{4}} K_{\frac{\nu}{2}}\!\left(\sqrt{-2t}\right)$
CF	$\frac{2}{\Gamma(\frac{\nu}{2})} \left(\frac{-it}{2}\right)^{\!\!\frac{\nu}{4}} K_{\frac{\nu}{2}}\!\left(\sqrt{-2it}\right)$

In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distribution) is 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 ν degrees of freedom, then according to the first definition, 1 / X has the inverse-chi-squared distribution with ν degrees of freedom; while according to the second definition, ν / X has the inverse-chi-squared distribution with ν degrees of freedom.

This distribution arises in Bayesian statistics.

It is a continuous distribution with a probability density function. The first definition yields a density function

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

The second definition yields a density function

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

In both cases, x > 0 and ν is the degrees of freedom parameter, and Γ is the gamma function. This article will deal with the first definition only. Both definitions are special cases of the scaled-inverse-chi-squared distribution. For the first definition σ² = 1 / ν and for the second definition σ² = 1.

Related distributions

• chi-squared: If X∼χ²(ν) and $Y = \frac{\nu}{X}$ then $Y ~ \sim \mbox{Inv-}\chi^2(\nu)$.
• Inverse gamma with $\alpha = \frac{\nu}{2}$ and $\beta = \frac{1}{2}$

See also

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

External links

• InvChisquare in geoR package for the R Language.