PlanetMath: multivariate gamma function (real-valued)

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multivariate gamma function (real-valued)

(Definition)

The real-valued multivariate gamma function is defined by

$\displaystyle \Gamma_m(a) = \int_{\mathfrak{S}} e^{-{\mathrm{Tr}}S} \left\vert S\right\vert^{a-{1 \over 2}(m+1)}\, {\rm d} S,$

(1)

where $\mathfrak{S}$ is the set of all $m \times m$ real, positive definite symmetric matrices, i.e.

$\displaystyle \mathfrak{S} = \left\{S \in \Bbb{R}^{m \times m} \mid S > 0, x^{\rm T}Sx > 0\, \forall\, x \in \mathbb{R}^{m \times 1}\setminus\{ 0\}\right\}.$

(2)

The real-valued multivariate gamma function can also be expressed in terms of the gamma function as follows

$\displaystyle \Gamma_m(a) = \pi^{{1 \over 4} m (m-1)} \prod\limits_{i=1}^{m}\Gamma\left(a-{1 \over 2}(i-1)\right).$

(3)

Reference

A. T. James, “Distributions of matrix variates and latent roots derived from normal samples,” Ann. Math. Statist., vol. 35, pp. 475-501, 1964.

"multivariate gamma function (real-valued)" is owned by rspuzio. [ full author list (2) | owner history (3) ]

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Also defines:

gamma function (multivariate real)

Keywords:

Gamma multivariate real

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Cross-references: terms, symmetric matrices, positive definite, real, gamma function

This is version 12 of multivariate gamma function (real-valued), born on 2004-05-13, modified 2006-12-01.
Object id is 5853, canonical name is MultivariateGammaFunctionRealValued.
Accessed 3270 times total.

Classification:

AMS MSC:

62H10 (Statistics :: Multivariate analysis :: Distribution of statistics)

Pending Errata and Addenda

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