Matrix gamma distribution

From Infogalactic: the planetary knowledge core
Jump to: navigation, search
Matrix gamma
Notation {\rm MG}_{p}(\alpha,\beta,\boldsymbol\Sigma)
Parameters 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-(p+1)/2} \exp\left({\rm tr}\left(-\frac{1}{\beta}\boldsymbol\Sigma^{-1}\mathbf{X}\right)\right)

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

This reduces to the Wishart distribution with \beta=2, \alpha=\frac{n}{2}.

See also

Notes

  1. 1.0 1.1 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

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