Matrix t-distribution

Matrix t
Notation	${\displaystyle {\rm {T}}_{n,p}(\nu ,\mathbf {M} ,{\boldsymbol {\Sigma }},{\boldsymbol {\Omega }})}$
Parameters	${\displaystyle \mathbf {M} }$ location (real ${\displaystyle n\times p}$ matrix) ${\displaystyle {\boldsymbol {\Omega }}}$ scale (positive-definite real ${\displaystyle n\times n}$ matrix) ${\displaystyle {\boldsymbol {\Sigma }}}$ scale (positive-definite real ${\displaystyle p\times p}$ matrix) ${\displaystyle \nu >0}$ degrees of freedom (real)
Support	${\displaystyle \mathbf {X} \in \mathbb {R} ^{n\times p}}$
PDF	${\displaystyle {\frac {\Gamma _{p}\left({\frac {\nu +n+p-1}{2}}\right)}{(\pi )^{\frac {np}{2}}\Gamma _{p}\left({\frac {\nu +p-1}{2}}\right)}}|{\boldsymbol {\Omega }}|^{-{\frac {n}{2}}}|{\boldsymbol {\Sigma }}|^{-{\frac {p}{2}}}}$ ${\displaystyle \times \left|\mathbf {I} _{p}+{\boldsymbol {\Sigma }}^{-1}(\mathbf {X} -\mathbf {M} ){\boldsymbol {\Omega }}^{-1}(\mathbf {X} -\mathbf {M} )^{\rm {T}}\right|^{-{\frac {\nu +n+p-1}{2}}}}$
CDF	No analytic expression
Mean	${\displaystyle \mathbf {M} }$ if ${\displaystyle \nu >1}$, else undefined
Mode	${\displaystyle \mathbf {M} }$
Variance	${\displaystyle \mathrm {cov} (\mathrm {vec} (\mathbf {X} ))={\frac {{\boldsymbol {\Sigma }}\otimes {\boldsymbol {\Omega }}}{\nu -2}}}$ if ${\displaystyle \nu >2}$, 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.¹²

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

For a matrix t-distribution, the probability density function at the point ${\displaystyle \mathbf {X} }$ of an ${\displaystyle n\times p}$ space is

${\displaystyle f(\mathbf {X} ;\nu ,\mathbf {M} ,{\boldsymbol {\Sigma }},{\boldsymbol {\Omega }})=K\times \left|\mathbf {I} _{n}+{\boldsymbol {\Sigma }}^{-1}(\mathbf {X} -\mathbf {M} ){\boldsymbol {\Omega }}^{-1}(\mathbf {X} -\mathbf {M} )^{\rm {T}}\right|^{-{\frac {\nu +n+p-1}{2}}},}$

where the constant of integration K is given by

${\displaystyle K={\frac {\Gamma _{p}\left({\frac {\nu +n+p-1}{2}}\right)}{(\pi )^{\frac {np}{2}}\Gamma _{p}\left({\frac {\nu +p-1}{2}}\right)}}|{\boldsymbol {\Omega }}|^{-{\frac {n}{2}}}|{\boldsymbol {\Sigma }}|^{-{\frac {p}{2}}}.}$

Here ${\displaystyle \Gamma _{p}}$ is the multivariate gamma function.

If ${\displaystyle \mathbf {X} \sim {\mathcal {T}}_{n\times p}(\nu ,\mathbf {M} ,\mathbf {\Sigma } ,\mathbf {\Omega } )}$, then we have the following properties:²

The mean, or expected value is, if ${\displaystyle \nu >1}$:

${\displaystyle E[\mathbf {X} ]=\mathbf {M} }$

and we have the following second-order expectations, if ${\displaystyle \nu >2}$:

${\displaystyle E[(\mathbf {X} -\mathbf {M} )(\mathbf {X} -\mathbf {M} )^{T}]={\frac {\mathbf {\Sigma } \operatorname {tr} (\mathbf {\Omega } )}{\nu -2}}}$

${\displaystyle E[(\mathbf {X} -\mathbf {M} )^{T}(\mathbf {X} -\mathbf {M} )]={\frac {\mathbf {\Omega } \operatorname {tr} (\mathbf {\Sigma } )}{\nu -2}}}$

where ${\displaystyle \operatorname {tr} }$ denotes trace.

More generally, for appropriately dimensioned matrices A,B,C:

${\displaystyle {\begin{aligned}E[(\mathbf {X} -\mathbf {M} )\mathbf {A} (\mathbf {X} -\mathbf {M} )^{T}]&={\frac {\mathbf {\Sigma } \operatorname {tr} (\mathbf {A} ^{T}\mathbf {\Omega } )}{\nu -2}}\\E[(\mathbf {X} -\mathbf {M} )^{T}\mathbf {B} (\mathbf {X} -\mathbf {M} )]&={\frac {\mathbf {\Omega } \operatorname {tr} (\mathbf {B} ^{T}\mathbf {\Sigma } )}{\nu -2}}\\E[(\mathbf {X} -\mathbf {M} )\mathbf {C} (\mathbf {X} -\mathbf {M} )]&={\frac {\mathbf {\Sigma } \mathbf {C} ^{T}\mathbf {\Omega } }{\nu -2}}\end{aligned}}}$

Transpose transform:

${\displaystyle \mathbf {X} ^{T}\sim {\mathcal {T}}_{p\times n}(\nu ,\mathbf {M} ^{T},\mathbf {\Omega } ,\mathbf {\Sigma } )}$

Linear transform: let A (r-by-n), be of full rank r ≤ n and B (p-by-s), be of full rank s ≤ p, then:

${\displaystyle \mathbf {AXB} \sim {\mathcal {T}}_{r\times s}(\nu ,\mathbf {AMB} ,\mathbf {A\Sigma A} ^{T},\mathbf {B} ^{T}\mathbf {\Omega B} )}$

The characteristic function and various other properties can be derived from the re-parameterised formulation (see below).

Re-parameterized matrix t
Notation	${\displaystyle {\rm {T}}_{n,p}(\alpha ,\beta ,\mathbf {M} ,{\boldsymbol {\Sigma }},{\boldsymbol {\Omega }})}$
Parameters	${\displaystyle \mathbf {M} }$ location (real ${\displaystyle n\times p}$ matrix) ${\displaystyle {\boldsymbol {\Omega }}}$ scale (positive-definite real ${\displaystyle p\times p}$ matrix) ${\displaystyle {\boldsymbol {\Sigma }}}$ scale (positive-definite real ${\displaystyle n\times n}$ matrix) ${\displaystyle \alpha >(p-1)/2}$ shape parameter ${\displaystyle \beta >0}$ scale parameter
Support	${\displaystyle \mathbf {X} \in \mathbb {R} ^{n\times p}}$
PDF	${\displaystyle {\frac {\Gamma _{p}(\alpha +n/2)}{(2\pi /\beta )^{\frac {np}{2}}\Gamma _{p}(\alpha )}}|{\boldsymbol {\Omega }}|^{-{\frac {n}{2}}}|{\boldsymbol {\Sigma }}|^{-{\frac {p}{2}}}}$ ${\displaystyle \times \left|\mathbf {I} _{n}+{\frac {\beta }{2}}{\boldsymbol {\Sigma }}^{-1}(\mathbf {X} -\mathbf {M} ){\boldsymbol {\Omega }}^{-1}(\mathbf {X} -\mathbf {M} )^{\rm {T}}\right|^{-(\alpha +n/2)}}$ ${\displaystyle \Gamma _{p}}$ is the multivariate gamma function.
CDF	No analytic expression
Mean	${\displaystyle \mathbf {M} }$ if ${\displaystyle \alpha >p/2}$, else undefined
Variance	${\displaystyle {\frac {2({\boldsymbol {\Sigma }}\otimes {\boldsymbol {\Omega }})}{\beta (2\alpha -p-1)}}}$ if ${\displaystyle \alpha >(p+1)/2}$, else undefined
CF	see below

An alternative parameterisation of the matrix t-distribution uses two parameters ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ in place of ${\displaystyle \nu }$.³

This formulation reduces to the standard matrix t-distribution with ${\displaystyle \beta =2,\alpha ={\frac {\nu +p-1}{2}}.}$

This formulation of the matrix t-distribution can be derived as the compound distribution that results from an infinite mixture of a matrix normal distribution with an inverse multivariate gamma distribution placed over either of its covariance matrices.

If ${\displaystyle \mathbf {X} \sim {\rm {T}}_{n,p}(\alpha ,\beta ,\mathbf {M} ,{\boldsymbol {\Sigma }},{\boldsymbol {\Omega }})}$ then²³

${\displaystyle \mathbf {X} ^{\rm {T}}\sim {\rm {T}}_{p,n}(\alpha ,\beta ,\mathbf {M} ^{\rm {T}},{\boldsymbol {\Omega }},{\boldsymbol {\Sigma }}).}$

The property above comes from Sylvester's determinant theorem:

${\displaystyle \det \left(\mathbf {I} _{n}+{\frac {\beta }{2}}{\boldsymbol {\Sigma }}^{-1}(\mathbf {X} -\mathbf {M} ){\boldsymbol {\Omega }}^{-1}(\mathbf {X} -\mathbf {M} )^{\rm {T}}\right)=}$

${\displaystyle \det \left(\mathbf {I} _{p}+{\frac {\beta }{2}}{\boldsymbol {\Omega }}^{-1}(\mathbf {X} ^{\rm {T}}-\mathbf {M} ^{\rm {T}}){\boldsymbol {\Sigma }}^{-1}(\mathbf {X} ^{\rm {T}}-\mathbf {M} ^{\rm {T}})^{\rm {T}}\right).}$

If ${\displaystyle \mathbf {X} \sim {\rm {T}}_{n,p}(\alpha ,\beta ,\mathbf {M} ,{\boldsymbol {\Sigma }},{\boldsymbol {\Omega }})}$ and ${\displaystyle \mathbf {A} (n\times n)}$ and ${\displaystyle \mathbf {B} (p\times p)}$ are nonsingular matrices then²³

${\displaystyle \mathbf {AXB} \sim {\rm {T}}_{n,p}(\alpha ,\beta ,\mathbf {AMB} ,\mathbf {A} {\boldsymbol {\Sigma }}\mathbf {A} ^{\rm {T}},\mathbf {B} ^{\rm {T}}{\boldsymbol {\Omega }}\mathbf {B} ).}$

The characteristic function is³

${\displaystyle \phi _{T}(\mathbf {Z} )={\frac {\exp({\rm {tr}}(i\mathbf {Z} '\mathbf {M} ))|{\boldsymbol {\Omega }}|^{\alpha }}{\Gamma _{p}(\alpha )(2\beta )^{\alpha p}}}|\mathbf {Z} '{\boldsymbol {\Sigma }}\mathbf {Z} |^{\alpha }B_{\alpha }\left({\frac {1}{2\beta }}\mathbf {Z} '{\boldsymbol {\Sigma }}\mathbf {Z} {\boldsymbol {\Omega }}\right),}$

where

${\displaystyle B_{\delta }(\mathbf {WZ} )=|\mathbf {W} |^{-\delta }\int _{\mathbf {S} >0}\exp \left({\rm {tr}}(-\mathbf {SW} -\mathbf {S^{-1}Z} )\right)|\mathbf {S} |^{-\delta -{\frac {1}{2}}(p+1)}d\mathbf {S} ,}$

and where ${\displaystyle B_{\delta }}$ is the type-two Bessel function of Herz of a matrix argument.