Inverse-chi-squared distribution

In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distribution^[1]) 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	${\displaystyle \nu >0\!}$
Support	${\displaystyle x\in (0,\infty )\!}$
PDF	${\displaystyle {\frac {2^{-\nu /2}}{\Gamma (\nu /2)}}\,x^{-\nu /2-1}e^{-1/(2x)}\!}$
CDF	${\displaystyle \Gamma \!\left({\frac {\nu }{2}},{\frac {1}{2x}}\right){\bigg /}\,\Gamma \!\left({\frac {\nu }{2}}\right)\!}$
Mean	${\displaystyle {\frac {1}{\nu -2}}\!}$ for ${\displaystyle \nu >2\!}$
Median	${\displaystyle \approx {\dfrac {1}{\nu {\bigg (}1-{\dfrac {2}{9\nu }}{\bigg )}^{3}}}}$
Mode	${\displaystyle {\frac {1}{\nu +2}}\!}$
Variance	${\displaystyle {\frac {2}{(\nu -2)^{2}(\nu -4)}}\!}$ 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	${\displaystyle {\frac {\nu }{2}}\!+\!\ln \!\left({\frac {\nu }{2}}\Gamma \!\left({\frac {\nu }{2}}\right)\right)}$ ${\displaystyle \!-\!\left(1\!+\!{\frac {\nu }{2}}\right)\psi \!\left({\frac {\nu }{2}}\right)}$
MGF	${\displaystyle {\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-t}{2i}}\right)^{\!\!{\frac {\nu }{4}}}K_{\frac {\nu }{2}}\!\left({\sqrt {-2t}}\right)}$; does not exist as real valued function
CF	${\displaystyle {\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-it}{2}}\right)^{\!\!{\frac {\nu }{4}}}K_{\frac {\nu }{2}}\!\left({\sqrt {-2it}}\right)}$

Definition

The inverse-chi-squared distribution (or inverted-chi-square distribution^[1] ) 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 ${\displaystyle X}$  has the chi-squared distribution with ${\displaystyle \nu }$  degrees of freedom, then according to the first definition, ${\displaystyle 1/X}$  has the inverse-chi-squared distribution with ${\displaystyle \nu }$  degrees of freedom; while according to the second definition, ${\displaystyle \nu /X}$  has the inverse-chi-squared distribution with ${\displaystyle \nu }$  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

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

while the second definition yields the density function

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

In both cases, ${\displaystyle x>0}$  and ${\displaystyle \nu }$  is the degrees of freedom parameter. Further, ${\displaystyle \Gamma }$  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 ${\displaystyle \sigma ^{2}=1/\nu ,}$  while for the second definition ${\displaystyle \sigma ^{2}=1}$ .

Related distributions

• 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

See also

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

References

1. Bernardo, J.M.; Smith, A.F.M. (1993) Bayesian Theory, Wiley (pages 119, 431) ISBN 0-471-49464-X

External links

• in geoR package for the R Language.