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.

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

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