# 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.[1] The matrix t-distribution shares the same relationship with the multivariate t-distribution that the matrix normal distribution shares with the multivariate normal distribution.[clarification needed] 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.[citation needed]

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

Notation ${\rm T}_{n,p}(\nu,\mathbf{M},\boldsymbol\Sigma, \boldsymbol\Omega)$ $\mathbf{M}$ 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) $\nu$ degrees of freedom $\mathbf{X} \in\mathbb{R}^{n\times p}$ $\frac{\Gamma_p\left(\frac{\nu+n+p-1}{2}\right)}{(\pi)^\frac{np}{2} \Gamma_p\left(\frac{\nu+p-1}{2}\right)} |\boldsymbol\Omega|^{-\frac{n}{2}} |\boldsymbol\Sigma|^{-\frac{p}{2}}$ $\times \left|\mathbf{I}_n + \boldsymbol\Sigma^{-1}(\mathbf{X} - \mathbf{M})\boldsymbol\Omega^{-1}(\mathbf{X}-\mathbf{M})^{\rm T}\right|^{-\frac{\nu+n+p-1}{2}}$ No analytic expression $\mathbf{M}$ if $\nu + p - n > 1$, else undefined $\mathbf{M}$ $\frac{\boldsymbol\Sigma \otimes \boldsymbol\Omega}{\nu-2}$ if $\nu > 2$, else undefined see below

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

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

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

Notation ${\rm T}_{n,p}(\alpha,\beta,\mathbf{M},\boldsymbol\Sigma, \boldsymbol\Omega)$ $\mathbf{M}$ 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 $\mathbf{X} \in\mathbb{R}^{n\times p}$ $\frac{\Gamma_p(\alpha+n/2)}{(2\pi/\beta)^\frac{np}{2} \Gamma_p(\alpha)} |\boldsymbol\Omega|^{-\frac{n}{2}} |\boldsymbol\Sigma|^{-\frac{p}{2}}$ $\times \left|\mathbf{I}_n + \frac{\beta}{2}\boldsymbol\Sigma^{-1}(\mathbf{X} - \mathbf{M})\boldsymbol\Omega^{-1}(\mathbf{X}-\mathbf{M})^{\rm T}\right|^{-(\alpha+n/2)}$ $\Gamma_p$ is the multivariate gamma function. No analytic expression $\mathbf{M}$ $\frac{2(\boldsymbol\Sigma \otimes \boldsymbol\Omega)}{\beta(2\alpha-n-1)}$ 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.[2]

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[citation needed]

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

This makes use of the following:[citation needed]

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

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[citation needed]

$\mathbf{AXB} \sim {\rm T}_{n,p}(\alpha,\beta,\mathbf{AMB},\mathbf{A}\boldsymbol\Sigma\mathbf{A}^{\rm T}, \mathbf{B}^{\rm T}\boldsymbol\Omega\mathbf{B}) .$
$\phi_T(\mathbf{Z}) = \frac{\exp({\rm tr}(i\mathbf{Z}'\mathbf{M}))|\boldsymbol\Omega|^\alpha}{\Gamma_p(\alpha)(2\beta)^{\alpha p}} |\mathbf{Z}'\boldsymbol\Sigma\mathbf{Z}|^\alpha B_\alpha\left(\frac{1}{2\beta}\mathbf{Z}'\boldsymbol\Sigma\mathbf{Z}\boldsymbol\Omega\right),$

where

$B_\delta(\mathbf{WZ}) = |\mathbf{W}|^{-\delta} \int_{\mathbf{S}>0} \exp\left({\rm tr}(-\mathbf{SW}-\mathbf{S^{-1}Z})\right)|\mathbf{S}|^{-\delta-\frac12(p+1)}d\mathbf{S},$

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