matrix gamma distribution

{{Short description|Generalization of gamma distribution}}

{{Probability distribution|

name =Matrix gamma|

type =density|

pdf_image =|

cdf_image =|

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)

cdf =|

mean =|

median =|

mode =|

variance =|

skewness =|

kurtosis =|

entropy =|

mgf =|

char =|

}}

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 \beta \boldsymbol\Sigma = 2 V, \alpha=\frac{n}{2}.

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

See also

Notes

{{Reflist}}

References

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

{{ProbDistributions|multivariate}}

Category:Random matrices

Category:Continuous distributions

Category:Multivariate continuous distributions

{{matrix-stub}}