Inverse chi-square distribution

The probability density function for the inverse chi-square distribution is

\[f(x, \nu) = \frac{2^{-\nu/2}}{\Gamma(\nu/2)} x^{-\nu/2 - 1} e^{-1/(2x)}\]

See the Wikipedia article “Inverse-chi-squared distribution” for more information. The functions here implement the first definition given in the wikipedia article. That is, if X has the chi-square distribution with \(\nu\) degrees of freedom, then 1/X has the inverse chi-square distribution with \(\nu\) degrees of freedom.

mpsci.distributions.invchi2.cdf(x, nu)

CDF for the inverse chi-square distribution.

mpsci.distributions.invchi2.logpdf(x, nu)

Logarithm of the PDF for the inverse chi-square distribution.

mpsci.distributions.invchi2.mean(nu)

Mean of the inverse chi-square distribution.

For \(\nu > 2\), the mean is \(1/(\nu - 2)\).

mpsci.distributions.invchi2.mode(nu)

Mode of the inverse chi-square distribution.

The mode is max(k - 2, 0).

mpsci.distributions.invchi2.pdf(x, nu)

PDF for the inverse chi-square distribution.

mpsci.distributions.invchi2.sf(x, nu)

Survival function for the inverse chi-square distribution.

mpsci.distributions.invchi2.support(nu)

Support of the inverse chi-square distribution.

mpsci.distributions.invchi2.var(nu)

Variance of the inverse chi-square distribution.

For \(\nu > 4\), the variance is

\[\frac{2}{(\nu - 2)^2 (\nu - 4)}\]