Matrix F-distribution

In statistics, the matrix F distribution is a matrix variate generalization of the F distribution which is defined on real-valued positive-definite matrices. In Bayesian statistics it can be used as the semi conjugate prior for the covariance matrix or precision matrix of multivariate normal distributions, and related distributions.

Density

The standard matrix F distribution, with an identity scale matrix, was originally derived by. When considering independent distributions,

and, and define, then.

This construction is useful to construct a semi-conjugate prior for a covariance matrix.

If and, then, after integrating out, has a matrix F-distribution, i.e.,

This construction is useful to construct a semi-conjugate prior for a precision matrix.

Suppose has a matrix F distribution. Partition the matrices and conformably with each other
where and are matrices, then we have.

Let.
The mean is given by:
The variance of elements of are given by:

The matrix F-distribution has also been termed the multivariate beta II distribution. See also, for a univariate version.
A univariate version of the matrix F distribution is the F-distribution. With and, and, the probability density function of the matrix F distribution becomes the univariate F distribution:
In the univariate case, with and, and when setting, then follows a half t distribution with scale parameter and degrees of freedom. The half t distribution is a common prior for standard deviations