Scale-inverse-chi-square distribution

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Moment-generating function (mgf)
Characteristic function

The scaled inverse chi-square distribution arises in Bayesian statistics. It is a more general distribution than the inverse-chi-square distribution. Its probability density function over the domain is

where is the degrees of freedom parameter and is the scale parameter. The cumulative distribution function is

where is the incomplete Gamma function, is the Gamma function and is a regularized Gamma function. The characteristic function is

where is the modified Bessel function of the second kind.

Parameter estimation

The maximum likelihood estimate of is

The maximum likelihood estimate of can be found using Newton's method on:

where is the digamma function. An initial estimate can be found by taking the formula for mean and solving it for Let be the sample mean. Then an initial estimate for is given by:

Related distributions

  • Relation to chi-square distribution: If and then
  • Relation to the inverse gamma distribution: If then .
  • The scale-inverse-chi-square distribution is a conjugate prior for the variance parameter of a normal distribution.

See also