Scale-inverse-chi-square distribution

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Scale-inverse-chi-square
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)
Probability density function (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}}
Cumulative distribution function (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\,
Median
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\,
Excess 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)
Moment-generating function (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)
Characteristic function \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-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 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

\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.

[edit] 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}.

[edit] Related distributions

[edit] See also


v • d • e
Probability distributions
 
Continuous univariate supported on a semi-infinite interval