Scaled Inverse Chi-squared Distribution

Characterization

The probability density function of the scaled inverse chi-squared distribution extends over the domain and is

$f(x; \nu, \tau^2)= \frac{(\tau^2\nu/2)^{\nu/2}}{\Gamma(\nu/2)}~ \frac{\exp\left}{x^{1+\nu/2}}$

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

$F(x; \nu, \tau^2)= \Gamma\left(\frac{\nu}{2},\frac{\tau^2\nu}{2x}\right) \left/\Gamma\left(\frac{\nu}{2}\right)\right.$

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.

Scaled Inverse Chi-squared Distribution - Characterization