Scale-inverse-chi-square distribution

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 $$x>0$$ is



f(x; \nu, \sigma^2)= \frac{(\sigma^2\nu/2)^{\nu/2}}{\Gamma(\nu/2)}~ \frac{\exp\left[ \frac{-\nu \sigma^2}{2 x}\right]}{x^{1+\nu/2}} $$

where $$\nu$$ is the degrees of freedom parameter and $$\sigma^2$$ is the scale parameter. The cumulative distribution function is


 * $$F(x; \nu, \sigma^2)=

\Gamma\left(\frac{\nu}{2},\frac{\sigma^2\nu}{2x}\right) \left/\Gamma\left(\frac{\nu}{2}\right)\right.$$
 * $$=Q\left(\frac{\nu}{2},\frac{\sigma^2\nu}{2x}\right)$$

where $$\Gamma(a,x)$$ is the incomplete Gamma function, $$\Gamma(x)$$ is the Gamma function and $$Q(a,x)$$ is a regularized Gamma function. The characteristic function is


 * $$\varphi(t;\nu,\sigma^2)=$$
 * $$\frac{2}{\Gamma(\frac{\nu}{2})}\left(\frac{-i\sigma^2\nu t}{2}\right)^{\!\!\frac{\nu}{4}}\!\!K_{\frac{\nu}{2}}\left(\sqrt{-2i\sigma^2\nu t}\right)$$

where $$K_{\frac{\nu}{2}}(z)$$ is the modified Bessel function of the second kind.

Parameter estimation
The maximum likelihood estimate of $$\sigma^2$$ is


 * $$\sigma^2 = n/\sum_{i=1}^N \frac{1}{x_i}.$$

The maximum likelihood estimate of $$\frac{\nu}{2}$$ can be found using Newton's method on:


 * $$\ln(\frac{\nu}{2}) + \psi(\frac{\nu}{2}) = \sum_{i=1}^N \ln(x_i) - n \ln(\sigma^2)$$

where $$\psi(x)$$ is the digamma function. An initial estimate can be found by taking the formula for mean and solving it for $$\nu.$$ Let $$\bar{x} = \frac{1}{n}\sum_{i=1}^N x_i$$ be the sample mean. Then an initial estimate for $$\nu$$ is given by:


 * $$\frac{\nu}{2} = \frac{\bar{x}}{\bar{x} - \sigma^2}.$$

Related distributions

 * Relation to chi-square distribution: If $$X \sim \chi^2(\nu)$$ and $$Y = \frac{\sigma^2 \nu}{X}$$ then $$Y \sim \mbox{Scale-inv-}\chi^2(\nu, \sigma^2)$$
 * Relation to the inverse gamma distribution: If $$X \sim \textrm{Inv-Gamma}\left(\frac{\nu}{2}, \frac{\nu \sigma^2}{2}\right)$$ then $$X \sim \mbox{Scale-inv-}\chi^2(\nu, \sigma^2)$$.
 * The scale-inverse-chi-square distribution is a conjugate prior for the variance parameter of a normal distribution.