Matrix gamma distribution

Matrix gamma

Notation	${\rm {MG}}_{p}(\alpha ,\beta ,{\boldsymbol {\Sigma }})$
Parameters	$\alpha >{\frac {p-1}{2}}$ shape parameter (real) $\beta >0$ scale parameter ${\boldsymbol {\Sigma }}$ scale (positive-definite real $p\times p$ matrix)
Support	$\mathbf {X}$ positive-definite real $p\times p$ matrix
PDF	${\frac {|{\boldsymbol {\Sigma }}|^{-\alpha }}{\beta ^{p\alpha }\,\Gamma _{p}(\alpha )}}|\mathbf {X} |^{\alpha -{\frac {p+1}{2}}}\exp \left({\rm {tr}}\left(-{\frac {1}{\beta }}{\boldsymbol {\Sigma }}^{-1}\mathbf {X} \right)\right)$ $\Gamma _{p}$ is the multivariate gamma function.

In statistics, a matrix gamma distribution is a generalization of the gamma distribution to positive-definite matrices. 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta \boldsymbol\Sigma = 2 V, \alpha=\frac{n}{2}.}

Notice that the parameters Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol\Sigma} are not identified; the density depends on these two parameters through the product Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta\boldsymbol\Sigma} .

See also

• inverse matrix gamma distribution.
• matrix normal distribution.
• matrix t-distribution.
• Wishart distribution.

References

• Gupta, A. K.; Nagar, D. K. (1999) Matrix Variate Distributions, Chapman and Hall/CRC ISBN 978-1584880462