Inverse-chi-squared distribution explained

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.

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.

If

X follows a chi-squared distribution with \nu degrees of freedom then 1 /X follows the inverse chi-squared distribution with \nu degrees of freedom.

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

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

In the above

x> 0 and \nu is the degrees of freedom parameter. Further, \Gamma is the gamma function.

The inverse chi-squared distribution is a special case of the inverse-gamma distribution. with shape parameter

\alpha = (\nu)/(2) and scale parameter \beta = (1)/(2) .

Related distributions

• chi-squared

If

X \thicksim \chi ^{2(\nu)} and Y = (1)/(X) , then Y \thicksim Inv- \chi ^{2(\nu)}

• scaled-inverse chi-squared

If

X \thicksim Scale-inv- \chi ^{2(\nu,} 1 /\nu) , then X \thicksim inv- \chi ^{2(\nu)}

• Inverse gamma with

\alpha = (\nu)/(2) and \beta = (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.

Notes and References

1. Bernardo, J.M.; Smith, A.F.M. (1993) Bayesian Theory, Wiley (pages 119, 431)