Scaled-inverse-chi-squared distribution

Scaled-inverse-chi-squared

Probability density function None uploaded yet
Cumulative distribution function None uploaded yet
Parameters	$\nu > 0\,$ $\sigma^2 > 0\,$
Support	$x \in (0, \infty)$
PDF	$\frac{(\sigma^2\nu/2)^{\nu/2}}{\Gamma(\nu/2)}~ \frac{\exp\left[ \frac{-\nu \sigma^2}{2 x}\right]}{x^{1+\nu/2}}$
CDF	$\Gamma\left(\frac{\nu}{2},\frac{\sigma^2\nu}{2x}\right) \left/\Gamma\left(\frac{\nu}{2}\right)\right.$
Mean	$\frac{\nu \sigma^2}{\nu-2}$ for $\nu >2\,$
Mode	$\frac{\nu \sigma^2}{\nu+2}$
Variance	$\frac{2 \nu^2 \sigma^4}{(\nu-2)^2 (\nu-4)}$for $\nu >4\,$
Skewness	$\frac{4}{\nu-6}\sqrt{2(\nu-4)}$for $\nu >6\,$
Ex. kurtosis	$\frac{12(5\nu-22)}{(\nu-6)(\nu-8)}$for $\nu >8\,$
Entropy	$\frac{\nu}{2} \!+\!\ln\left(\frac{\sigma^2\nu}{2}\Gamma\left(\frac{\nu}{2}\right)\right)$ $\!-\!\left(1\!+\!\frac{\nu}{2}\right)\psi\left(\frac{\nu}{2}\right)$
MGF	$\frac{2}{\Gamma(\frac{\nu}{2})}\left(\frac{-\sigma^2\nu t}{2}\right)^{\!\!\frac{\nu}{4}}\!\!K_{\frac{\nu}{2}}\left(\sqrt{-2\sigma^2\nu t}\right)$
CF	$\frac{2}{\Gamma(\frac{\nu}{2})}\left(\frac{-i\sigma^2\nu t}{2}\right)^{\!\!\frac{\nu}{4}}\!\!K_{\frac{\nu}{2}}\left(\sqrt{-2i\sigma^2\nu t}\right)$

The scaled inverse chi-squared distribution arises in Bayesian statistics. The family of scaled inverse chi-squared distributions contains an extra scaling parameter compared to the inverse-chi-squared distribution, and it is essentially the same family of distributions as the inverse gamma distribution, but using a parametrization that may be more convenient for Bayesian statistics. Specifically, the scaled inverse chi-squared distribution can be used as a conjugate prior for the variance parameter of a normal distribution; however, it is more common to use the inverse gamma distribution formulation instead. This distribution is the maximum entropy distribution for a fixed first inverse moment (E(1 / X)) and first logarithmic moment (E(ln(X)).

Characterization

The probability density function of the scaled inverse chi-squared distribution extends over the domain x > 0 and 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 ν is the degrees of freedom parameter and σ² 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 Γ(a,x) is the incomplete Gamma function, Γ(x) is the Gamma function and Q(a,x) is a regularized Gamma function. The characteristic function is

φ(t;ν,σ²) =

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

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

Parameter estimation

The maximum likelihood estimate of σ² 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 ψ(x) is the digamma function. An initial estimate can be found by taking the formula for mean and solving it for ν. Let $\bar{x} = \frac{1}{n}\sum_{i=1}^N x_i$ be the sample mean. Then an initial estimate for ν is given by:

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

Related distributions

• If X∼Scale-inv-χ²(ν,σ²) then $k X \sim \mbox{Scale-inv-}\chi^2(\nu, k \sigma^2)\,$
• If $X \sim \mbox{inv-}\chi^2(\nu) \,$ (Inverse-chi-squared distribution) then $X \sim \mbox{Scale-inv-}\chi^2(\nu, 1/\nu) \,$
• If X∼Scale-inv-χ²(ν,σ²) then $\frac{X}{\sigma^2 \nu} \sim \mbox{inv-}\chi^2(\nu) \,$ (Inverse-chi-squared distribution)
• If X∼Scale-inv-χ²(ν,σ²) then $X \sim \textrm{Inv-Gamma}\left(\frac{\nu}{2}, \frac{\nu\sigma^2}{2}\right)$ (Inverse-gamma distribution)
• Scaled inverse chi square distribution is a special case of type 5 Pearson distribution