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: If the matrix has only one row, or only one column, the distributions become equivalent to the corresponding 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 of an space is
where the constant of integration K is given by
Here is the multivariate gamma function.

Properties

If, then we have the following properties:

Expected values

The mean, or expected value is, if :
and we have the following second-order expectations, if :
where denotes trace.
More generally, for appropriately dimensioned matrices A,'B,C':

Transformation

Transpose transform:
Linear transform: let A, be of full rank r ≤ n and B, be of full rank s ≤ p, then:
The characteristic function and various other properties can be derived from the re-parameterised formulation.

Re-parameterized matrix ''t''-distribution

An alternative parameterisation of the matrix t-distribution uses two parameters and in place of.
This formulation reduces to the standard matrix t-distribution with
This formulation of the matrix t-distribution can be derived as 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.