Matrix t-distribution

Matrix t
Notation
Parameters	location (real matrix); scale (positive-definite real matrix); scale (positive-definite real matrix) ; degrees of freedom (real)
Support
PDF
CDF	No analytic expression
Mean	if , else undefined
Mode
Variance	if , else undefined
CF	see below

In statistics, the matrix t-distribution (or matrix variate t-distribution) is the generalization of the multivariate t-distribution from vectors to matrices.^[1]^[2]

The matrix t-distribution shares the same relationship with the multivariate t-distribution that the matrix normal distribution shares with the multivariate normal distribution: If the matrix has only one row, or only one column, the distributions become equivalent to the corresponding (vector-)multivariate distribution. The matrix t-distribution is the compound distribution that results from an infinite mixture of a matrix normal distribution with an inverse Wishart distribution placed over either of its covariance matrices,^[1] and the multivariate t-distribution can be generated in a similar way.^[2]

In a Bayesian analysis of a multivariate linear regression model based on the matrix normal distribution, the matrix t-distribution is the posterior predictive distribution.^[3]

^ ^a ^b Zhu, Shenghuo and Kai Yu and Yihong Gong (2007). "Predictive Matrix-Variate t Models." In J. C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, NIPS '07: Advances in Neural Information Processing Systems 20, pages 1721–1728. MIT Press, Cambridge, MA, 2008. The notation is changed a bit in this article for consistency with the matrix normal distribution article.
^ ^a ^b Gupta, Arjun K and Nagar, Daya K (1999). Matrix variate distributions. CRC Press. pp. Chapter 4.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ Cite error: The named reference Iranmanesh was invoked but never defined (see the help page).

[Zhu-1] Zhu, Shenghuo and Kai Yu and Yihong Gong (2007). "Predictive Matrix-Variate t Models." In J. C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, NIPS '07: Advances in Neural Information Processing Systems 20, pages 1721–1728. MIT Press, Cambridge, MA, 2008. The notation is changed a bit in this article for consistency with the matrix normal distribution article.

[Gupta-2] Gupta, Arjun K and Nagar, Daya K (1999). Matrix variate distributions. CRC Press. pp. Chapter 4.{{cite book}}: CS1 maint: multiple names: authors list (link)

[Iranmanesh-3] Cite error: The named reference Iranmanesh was invoked but never defined (see the help page).

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Notation	${\rm {T}}_{n,p}(\nu ,\mathbf {M} ,{\boldsymbol {\Sigma }},{\boldsymbol {\Omega }})$
Parameters	$\mathbf {M}$ location (real $n\times p$ matrix) ${\boldsymbol {\Omega }}$ scale (positive-definite real $p\times p$ matrix) ${\boldsymbol {\Sigma }}$ scale (positive-definite real $n\times n$ matrix) $\nu >0$ degrees of freedom (real)
Support	$\mathbf {X} \in \mathbb {R} ^{n\times p}$
PDF	${\frac {\Gamma _{p}\left({\frac {\nu +n+p-1}{2}}\right)}{(\pi )^{\frac {np}{2}}\Gamma _{p}\left({\frac {\nu +p-1}{2}}\right)}}\|{\boldsymbol {\Omega }}\|^{-{\frac {n}{2}}}\|{\boldsymbol {\Sigma }}\|^{-{\frac {p}{2}}}$ $\times \left\|\mathbf {I} _{n}+{\boldsymbol {\Sigma }}^{-1}(\mathbf {X} -\mathbf {M} ){\boldsymbol {\Omega }}^{-1}(\mathbf {X} -\mathbf {M} )^{\rm {T}}\right\|^{-{\frac {\nu +n+p-1}{2}}}$
CDF	No analytic expression
Mean	$\mathbf {M}$ if $\nu >1$ , else undefined
Mode	$\mathbf {M}$
Variance	$\mathrm {cov} (\mathrm {vec} (\mathbf {X} ))={\frac {{\boldsymbol {\Sigma }}\otimes {\boldsymbol {\Omega }}}{\nu -2}}$ if $\nu >2$ , else undefined
CF	see below