Matrix t-distribution

Matrix t
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.

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, and the multivariate t-distribution can be generated in a similar way.

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.

Matrix t
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 $n\times n$ matrix) ${\boldsymbol {\Sigma }}$ scale (positive-definite real $p\times p$ matrix) $\nu >0$ degrees of freedom (real)
Support	$\mathbf {X} \in \mathbb {R} ^{n\times p}$
PDF	${\frac {\Gamma _{p}\left({\frac {\nu +n+p-1}{2}}\right)}{(\pi )^{\frac {np}{2}}\Gamma _{p}\left({\frac {\nu +p-1}{2}}\right)}}\|{\boldsymbol {\Omega }}\|^{-{\frac {n}{2}}}\|{\boldsymbol {\Sigma }}\|^{-{\frac {p}{2}}}$ $\times \left\|\mathbf {I} _{p}+{\boldsymbol {\Sigma }}^{-1}(\mathbf {X} -\mathbf {M} ){\boldsymbol {\Omega }}^{-1}(\mathbf {X} -\mathbf {M} )^{\rm {T}}\right\|^{-{\frac {\nu +n+p-1}{2}}}$
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