Matrix gamma distribution

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

In statistics, a matrix gamma distribution is a generalization of the gamma distribution to positive-definite matrices.^[1] 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.^[1]

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

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

Notes

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^ ^a ^b Iranmanesh, Anis, M. Arashib and S. M. M. Tabatabaey (2010). "On Conditional Applications of Matrix Variate Normal Distribution". Iranian Journal of Mathematical Sciences and Informatics, 5:2, pp. 33–43.

References

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

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Probability distributions (list)

Discrete
univariate

with finite support	Benford Bernoulli Beta-binomial Binomial Categorical Hypergeometric Negative Poisson binomial Rademacher Soliton Discrete uniform Zipf Zipf–Mandelbrot
with infinite support	Beta negative binomial Borel Conway–Maxwell–Poisson Discrete phase-type Delaporte Extended negative binomial Flory–Schulz Gauss–Kuzmin Geometric Logarithmic Mixed Poisson Negative binomial Panjer Parabolic fractal Poisson Skellam Yule–Simon Zeta

Continuous
univariate

supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Generalized Beta rectangular Continuous Bernoulli Irwin–Hall Kumaraswamy Logit-normal Noncentral beta PERT Raised cosine Reciprocal Triangular U-quadratic Uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr Chi Chi-squared Noncentral Inverse Scaled Dagum Davis Erlang Hyper Exponential Hyperexponential Hypoexponential Logarithmic F Noncentral Folded normal Fréchet Gamma Generalized Inverse gamma/Gompertz Gompertz Shifted Half-logistic Half-normal Hotelling's T-squared Inverse Gaussian Generalized Kolmogorov Lévy Log-Cauchy Log-Laplace Log-logistic Log-normal Log-t Lomax Matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto Phase-type Poly-Weibull Rayleigh Relativistic Breit–Wigner Rice Truncated normal type-2 Gumbel Weibull Discrete Wilks's lambda
supported on the whole real line	Cauchy Exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q Generalized normal Generalized hyperbolic Geometric stable Gumbel Holtsmark Hyperbolic secant Johnson's S_U Landau Laplace Asymmetric Logistic Noncentral t Normal (Gaussian) Normal-inverse Gaussian Skew normal Slash Stable Student's t Tracy–Widom Variance-gamma Voigt
with support whose type varies	Generalized chi-squared Generalized extreme value Generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull Shifted log-logistic Tukey lambda