Matrix t-distribution

Matrix t

Notation	${\mathrm{T}}_{n,p}(\nu ,\mathbf{𝐌},\mathrm{\Sigma },\mathrm{\Omega })$
Parameters	$\mathbf{𝐌}$ location (real $n\times p$ matrix) $\mathrm{\Omega }$ scale (positive-definite real $p\times p$ matrix) $\mathrm{\Sigma }$ scale (positive-definite real $n\times n$ matrix) $\nu$ degrees of freedom
Support	$\mathbf{𝐗}\in {\mathbb{ℝ}}^{n\times p}$
PDF	$\frac{{\mathrm{\Gamma }}_p\left(\frac{\nu +n+p-1}{2}\right)}{(\pi {)}^{\frac{np}{2}}{\mathrm{\Gamma }}_p\left(\frac{\nu +p-1}{2}\right)}|\mathrm{\Omega }{|}^{-\frac{n}{2}}|\mathrm{\Sigma }{|}^{-\frac{p}{2}}$ $\times {|{\mathbf{𝐈}}_n+{\mathrm{\Sigma }}^{-1}(\mathbf{𝐗}-\mathbf{𝐌}){\mathrm{\Omega }}^{-1}(\mathbf{𝐗}-\mathbf{𝐌}{)}^{\mathrm{T}}|}^{-\frac{\nu +n+p-1}{2}}$
CDF	No analytic expression
Mean	$\mathbf{𝐌}$ if $\nu +p-n>1$, else undefined
Mode	$\mathbf{𝐌}$
Variance	$\frac{\mathrm{\Sigma }\otimes \mathrm{\Omega }}{\nu -2}$ if $\nu >2$, 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. The matrix t-distribution shares the same relationship with the multivariate t-distribution that the matrix normal distribution shares with the multivariate normal distribution. For example, the matrix t-distribution is the compound distribution that results from sampling from a matrix normal distribution having sampled the covariance matrix of the matrix normal from an inverse Wishart distribution.

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

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

where the constant of integration K is given by

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

Here ${\mathrm{\Gamma }}_p$ 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	${\mathrm{T}}_{n,p}(\alpha ,\beta ,\mathbf{𝐌},\mathrm{\Sigma },\mathrm{\Omega })$
Parameters	$\mathbf{𝐌}$ location (real $n\times p$ matrix) $\mathrm{\Omega }$ scale (positive-definite real $p\times p$ matrix) $\mathrm{\Sigma }$ 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	$\frac{{\mathrm{\Gamma }}_p(\alpha +n/2)}{(2\pi /\beta {)}^{\frac{np}{2}}{\mathrm{\Gamma }}_p(\alpha )}|\mathrm{\Omega }{|}^{-\frac{n}{2}}|\mathrm{\Sigma }{|}^{-\frac{p}{2}}$ $\times {|{\mathbf{𝐈}}_n+\frac{\beta }{2}{\mathrm{\Sigma }}^{-1}(\mathbf{𝐗}-\mathbf{𝐌}){\mathrm{\Omega }}^{-1}(\mathbf{𝐗}-\mathbf{𝐌}{)}^{\mathrm{T}}|}^{-(\alpha +n/2)}$ ${\mathrm{\Gamma }}_p$ is the multivariate gamma function.
CDF	No analytic expression
Mean	$\mathbf{𝐌}$
Variance	$\frac{2(\mathrm{\Sigma }\otimes \mathrm{\Omega })}{\beta (2\alpha -p-1)}$
CF	see below

The generalized matrix t-distribution is a generalization of the matrix t-distribution with two parameters α and β in place of ν.

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

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

The property above comes from Sylvester's determinant theorem:

$\det ({\mathbf{𝐈}}_n+\frac{\beta }{2}{\mathrm{\Sigma }}^{-1}(\mathbf{𝐗}-\mathbf{𝐌}){\mathrm{\Omega }}^{-1}(\mathbf{𝐗}-\mathbf{𝐌}{)}^{\mathrm{T}})=$

$\det ({\mathbf{𝐈}}_p+\frac{\beta }{2}{\mathrm{\Omega }}^{-1}({\mathbf{𝐗}}^{\mathrm{T}}-{\mathbf{𝐌}}^{\mathrm{T}}){\mathrm{\Sigma }}^{-1}({\mathbf{𝐗}}^{\mathrm{T}}-{\mathbf{𝐌}}^{\mathrm{T}}{)}^{\mathrm{T}}).$

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

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

The characteristic function is

${\varphi }_T(\mathbf{𝐙})=\frac{\exp (\mathrm{t}\mathrm{r}(i{\mathbf{𝐙}}^{\prime }\mathbf{𝐌}))|\mathrm{\Omega }{|}^{\alpha }}{{\mathrm{\Gamma }}_p(\alpha )(2\beta {)}^{\alpha p}}|{\mathbf{𝐙}}^{\prime }\mathrm{\Sigma }\mathbf{𝐙}{|}^{\alpha }{B}_{\alpha }\left(\frac{1}{2\beta }{\mathbf{𝐙}}^{\prime }\mathrm{\Sigma }\mathbf{𝐙}\mathrm{\Omega }\right),$

where

$B_{\delta }(\mathbf{𝐖}\mathbf{𝐙})=|\mathbf{𝐖}{|}^{-\delta }\underset{\mathbf{𝐒}>0}{\int }\exp (\mathrm{t}\mathrm{r}(-\mathbf{𝐒}\mathbf{𝐖}-{\mathbf{𝐒}}^{\mathbf{-}\mathbf{𝟏}}\mathbf{𝐙}))|\mathbf{𝐒}{|}^{-\delta -\frac{1}{2}(p+1)}d\mathbf{𝐒},$

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