Multivariate Gamma Function

In mathematics, the multivariate Gamma distribution, \Gamma_p(\cdot), is a generalization of the Gamma function. It is useful in multivariate statistics. It has two equivalent definitions:
\Gamma_p(a)= \int_{S\in {\mathbf S}} \exp\left( -{\rm trace}(S)\right) \left|S\right|^{a-(p+1)/2} dS where {\mathbf S} is the set of all positive-definite matrices. The other is more useful in practice:
\Gamma_p(a)= \pi^{p(p-1)/4}\Pi_{j=1}^p \Gamma\lefta+(1-j)/2\right. Thus
  • \Gamma_1(a)=\Gamma(a)
  • \Gamma_2(a)=\pi^{1/2}\Gamma(a)\Gamma(a-1/2)
  • \Gamma_3(a)=\pi^{3/2}\Gamma(a)\Gamma(a-1/2)\Gamma(a-1)
and so on.

 

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