Matrix t-distribution

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

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 $p\times p$ 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	${\frac {{\boldsymbol {\Sigma }}\otimes {\boldsymbol {\Omega }}}{\nu -2}}$ if $\nu >2$, else undefined
CF	see below

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.

Definition

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

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

where the constant of integration K is given by

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

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

The characteristic function and various other properties can be derived from the generalized matrix t-distribution (see below).

Generalized matrix t-distribution

Generalized matrix t

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

The generalized matrix t-distribution is a generalization of the matrix t-distribution with two parameters $\alpha$ and $\beta$ in place of $\nu$.

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

The generalized matrix t-distribution is 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.

Properties

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

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

The property above comes from Sylvester's determinant theorem:

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

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

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

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

The characteristic function is

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

where

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

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

See also

• Multivariate t-distribution
• Matrix normal distribution

External links

• A C++ library for random matrix generator