Scaled-inverse-chi-squared distribution

Scaled-inverse-chi-squared
Probability density function
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Cumulative distribution function
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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%2B\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%2B2}
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} \!%2B\!\ln\left(\frac{\sigma^2\nu}{2}\Gamma\left(\frac{\nu}{2}\right)\right)

\!-\!\left(1\!%2B\!\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%2B\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}) %2B \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}.

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