Matrix t-distribution

Matrix t

Parameters	$\mathbf {M}$ location (real $p\times m$ matrix) ${\boldsymbol {\Omega }}$ rescaling matrix (positive-definite real $p\times p$ matrix) ${\boldsymbol {\Sigma }}$ scale matrix (positive-definite real $m\times m$ matrix) ν is the degree of freedom
Support	$\mathbf {X} \in \mathbb {R} ^{p\times m}$
PDF	${\frac {\Gamma _{p}\left({\frac {\nu +m+p-1}{2}}\right)}{(\nu \pi )^{\frac {mp}{2}}\Gamma _{p}\left({\frac {\nu +p-1}{2}}\right)}}|{\boldsymbol {\Omega }}|^{-{\frac {m}{2}}}|{\boldsymbol {\Sigma }}|^{-{\frac {p}{2}}}|\mathbf {I} _{p}+{\boldsymbol {\Omega }}^{-1}(\mathbf {X} -\mathbf {M} ){\boldsymbol {\Sigma }}^{-1}(\mathbf {X} -\mathbf {M} )^{\rm {T}}|^{-{\frac {\nu +m+p-1}{2}}}$
CDF	No analytic expression
Mean	if $n>1$, $\mathbf {M}$ else undefined
Median	$\mathbf {M}$
Mode	$\mathbf {M}$
Variance	if $n>2$, ${\frac {n}{n-2}}\mathbf {\Sigma }$ else undefined
Skewness	0

In statistics, a matrix t-distribution (or matrix Student distribution) is a generalization of the multivariate t-distribution to a matrix. The multivariate t-distribution is often defined as the compound distribution that results from infinite mixture of a multivariate normal distribution with the conjugate prior distribution over the variance (i.e. an inverse Wishart distribution). The matrix t-distribution results from a similar compound distribution based on a matrix normal distribution.

See also

• multivariate t-distribution.
• matrix normal distribution.

References

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