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. 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 $\mathbf$ of an $n\times p$ space is

:$f(\mathbf ; \nu,\mathbf,\boldsymbol\Sigma, \boldsymbol\Omega) = K \times \left, \mathbf_n + \boldsymbol\Sigma^(\mathbf - \mathbf)\boldsymbol\Omega^(\mathbf-\mathbf)^\^,$

where the constant of integration ''K'' is given by
:$K = \frac , \boldsymbol\Omega, ^ , \boldsymbol\Sigma, ^.$

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 ''α'' and ''β'' in place of ''ν''.Iranmanesh, Anis, M. Arashi and S. M. M. Tabatabaey (2010)
"On Conditional Applications of Matrix Variate Normal Distribution"
''Iranian Journal of Mathematical Sciences and Informatics'', 5:2, pp. 33–43.

This reduces to the standard matrix ''t''-distribution with $\beta=2, \alpha=\frac.$

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 $\mathbf \sim _(\alpha,\beta,\mathbf,\boldsymbol\Sigma, \boldsymbol\Omega)$ then
:$\mathbf^ \sim _(\alpha,\beta,\mathbf^,\boldsymbol\Omega, \boldsymbol\Sigma).$

The property above comes from Sylvester's determinant theorem:

:$\det\left(\mathbf_n + \frac\boldsymbol\Sigma^(\mathbf - \mathbf)\boldsymbol\Omega^(\mathbf-\mathbf)^\right) =$
::$\det\left(\mathbf_p + \frac\boldsymbol\Omega^(\mathbf^ - \mathbf^)\boldsymbol\Sigma^(\mathbf^-\mathbf^)^\right) .$

If $\mathbf \sim _(\alpha,\beta,\mathbf,\boldsymbol\Sigma, \boldsymbol\Omega)$ and $\mathbf(n\times n)$ and $\mathbf(p\times p)$ are nonsingular matrices then

:$\mathbf \sim _(\alpha,\beta,\mathbf,\mathbf\boldsymbol\Sigma\mathbf^, \mathbf^\boldsymbol\Omega\mathbf) .$

The characteristic function is

:$\phi_T(\mathbf) = \frac , \mathbf'\boldsymbol\Sigma\mathbf, ^\alpha B_\alpha\left(\frac\mathbf'\boldsymbol\Sigma\mathbf\boldsymbol\Omega\right),$

where
:$B_\delta(\mathbf) = , \mathbf, ^ \int_ \exp\left((-\mathbf-\mathbf)\right), \mathbf, ^d\mathbf,$

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

See also

* Multivariate ''t''-distribution
* Matrix normal distribution

Notes

External links

A C++ library for random matrix generator

{{ProbDistributions, multivariate

Random matrices
Multivariate continuous distributions