A.7 Log chi-squared distribution

X has a log chi-squared distribution on ν degrees of freedom, denoted

if X=log W where , or equivalently if X has density

(note that unlike itself this is a distribution over the whole line).

Because the logarithm of an variable differs from a log chi-squared variable simply by an additive constant, it is not necessary to consider such variables in any detail.

By considering the tth moment of a variable, it is easily shown that the moment generating function of a log chi-squared variable is

Writing

for the so-called digamma function, it follows that the mean and variance are

or (using Stirling’s approximation and its derivatives) approximately

The mode is