Multivariate gamma function - Definition

In mathematics, the multivariate gamma distribution , <math>\Gamma_p(\cdot)<math>, is a generalization of the gamma function. It is useful in multivariate statistics.

It has two equivalent definitions:

<math>

\Gamma_p(a)= \int_{S\in {\mathbf S}} \exp\left( -{\rm trace}(S)\right) \left|S\right|^{a-(p+1)/2} dS <math> where <math>{\mathbf S}<math> is the set of all positive-definite matrices. The other is more useful in practice:

<math>

\Gamma_p(a)= \pi^{p(p-1)/4}\Pi_{j=1}^p \Gamma\left[ a+(1-j)/2\right]. <math>

Thus

• <math>\Gamma_1(a)=\Gamma(a)<math>
• <math>\Gamma_2(a)=\pi^{1/2}\Gamma(a)\Gamma(a-1/2)<math>
• <math>\Gamma_3(a)=\pi^{3/2}\Gamma(a)\Gamma(a-1/2)\Gamma(a-1)<math>

and so on.