Multivariate gamma function: Difference between revisions

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In [[mathematics]], the '''multivariate gamma function''' Γ_''p'' is a generalization of the [[gamma function]]. It is useful in [[multivariate statistics]], appearing in the [[probability density function]] of the [[Wishart distribution|Wishart]] and [[inverse Wishart distribution]]s, and the [[matrix variate beta distribution]].<ref>{{Cite journal|last=James|first=Alan T.|date=June 1964-06|title=Distributions of Matrix Variates and Latent Roots Derived from Normal Samples|url=http://projecteuclid.org/euclid.aoms/1177703550|journal=The Annals of Mathematical Statistics|language=en|volume=35|issue=2|pages=475–501|doi=10.1214/aoms/1177703550|issn=0003-4851}}</ref>

It has two equivalent definitions. One is given as the following integral over the <math>p \times p</math> [[positive-definite matrix|positive-definite]] real matrices:

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and so on.

This can also be extended to non-integer values of p with the expression:

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</math>

{{nomore footnotes|date=May 2012}}<br />

==References==

{{Reflist}}

* 1. {{cite journal

|title=Distributions of Matrix Variates and Latent Roots Derived from Normal Samples

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|doi-access=free }}

* 2. A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall.

[[Category:Gamma and related functions]]