matrix gamma distribution

{{Short description|Generalization of gamma distribution}}

{{Probability distribution|

name =Matrix gamma|

type =density|

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notation =$\{\backslash rm\; MG\}\_\{p\}(\backslash alpha,\backslash beta,\backslash boldsymbol\backslash Sigma)$|

parameters = $\backslash alpha\; >\; \backslash frac\{p-1\}\{2\}$ shape parameter (real)

$\backslash beta\; >\; 0$ scale parameter

$\backslash boldsymbol\backslash Sigma$ scale (positive-definite real $p\backslash times\; p$ matrix)

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support =$\backslash mathbf\{X\}$ positive-definite real $p\backslash times\; p$ matrix|

pdf =$\backslash frac\{|\backslash boldsymbol\backslash Sigma|^\{-\backslash alpha\}\}\{\backslash beta^\{p\backslash alpha\}\backslash ,\backslash Gamma\_p(\backslash alpha)\}\; |\backslash mathbf\{X\}|^\{\backslash alpha-\backslash frac\{p+1\}\{2\}\}\; \backslash exp\backslash left(\{\backslash rm\; tr\}\backslash left(-\backslash frac\{1\}\{\backslash beta\}\backslash boldsymbol\backslash Sigma^\{-1\}\backslash mathbf\{X\}\backslash right)\backslash right)$

• $\backslash Gamma\_p$ is the multivariate gamma function.|

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In statistics, a matrix gamma distribution is a generalization of the gamma distribution to positive-definite matrices.Iranmanesh, Anis, M. Arashib and S. M. M. Tabatabaey (2010). [http://www.ijmsi.ir/browse.php?a_code=A-10-1-83&slc_lang=en&sid=1 "On Conditional Applications of Matrix Variate Normal Distribution"]. Iranian Journal of Mathematical Sciences and Informatics, 5:2, pp. 33–43. It is effectively a different parametrization of the Wishart distribution, and is used similarly, e.g. as the conjugate prior of the precision matrix of a multivariate normal distribution and matrix normal distribution. The compound distribution resulting from compounding a matrix normal with a matrix gamma prior over the precision matrix is a generalized matrix t-distribution.

A matrix gamma distributions is identical to a Wishart distribution with $\backslash beta\; \backslash boldsymbol\backslash Sigma\; =\; 2\; V,\; \backslash alpha=\backslash frac\{n\}\{2\}.$

Notice that the parameters $\backslash beta$ and $\backslash boldsymbol\backslash Sigma$ are not identified; the density depends on these two parameters through the product $\backslash beta\backslash boldsymbol\backslash Sigma$.