View source for Matrix gamma distribution

{{Probability distribution|
  name       =Matrix gamma|
  type       =density|
  pdf_image  =|
  cdf_image  =|
  notation   =<math>{\rm MG}_{p}(\alpha,\beta,\boldsymbol\Sigma)</math>|
  parameters = <math>\alpha > \frac{p-1}{2} </math> [[shape parameter]] ([[real number|real]])<br/>
<math>\beta > 0 </math> [[scale parameter]]<br/>
<math>\boldsymbol\Sigma</math> [[scale matrix|scale]] ([[positive-definite matrix|positive-definite]] real <math>p\times p</math> [[matrix (mathematics)|matrix]])
|
  support    =<math>\mathbf{X}</math> [[positive-definite matrix|positive-definite]] real <math>p\times p</math> [[matrix (mathematics)|matrix]]|
  pdf        =<math>\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)</math>
*<math>\Gamma_p</math> is the [[multivariate gamma function]].|
  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 matrix|positive-definite matrices]].<ref name="iranmanesh">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.</ref> 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]].<ref name="iranmanesh"/>

A matrix gamma distributions is identical to a [[Wishart distribution]] with <math>\beta \boldsymbol\Sigma = 2 V, \alpha=\frac{n}{2}.</math>

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

== See also ==
* [[inverse matrix gamma distribution]].
* [[matrix normal distribution]].
* [[matrix t-distribution]].
* [[Wishart distribution]].

==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]]


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