In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distributionBernardo, J.M.; Smith, A.F.M. (1993) Bayesian Theory, Wiley (pages 119, 431) ) 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.
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.
If follows a chi-squared distribution with degrees of freedom then follows the inverse chi-squared distribution with degrees of freedom.
The probability density function of the inverse chi-squared distribution is given by
In the above and is the degrees of freedom parameter. Further, is the gamma function.
The inverse chi-squared distribution is a special case of the inverse-gamma distribution. with shape parameter and scale parameter .
Related distributions
chi-squared: If and , then scaled-inverse chi-squared: If , then Inverse gamma with and Inverse chi-squared distribution is a special case of type 5 Pearson distribution
See also Scaled-inverse-chi-squared distribution Inverse-Wishart distribution
References
External links InvChisquare in geoR package for the R Language.