Matrix t-distribution


In statistics, the matrix 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 of an space is
where the constant of integration K is given by
Here is the multivariate gamma function.
The characteristic function and various other properties can be derived from the generalized matrix t-distribution.

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.
This reduces to the standard matrix t-distribution with
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 then
The property above comes from Sylvester's determinant theorem:
If and and are nonsingular matrices then
The characteristic function is
where
and where is the type-two Bessel function of Herz of a matrix argument.