The Wishart distribution, spherical polynomials and the symmetric group.

Helene Massam
York University

Abstract: The Wishart distribution is the distribution of the estimate of the
covariance parameter in the multivariate Gaussian model.
In many statistical problems, the test statistic is a fairly
complicated function of this estimate and it is impossible
to find its exact distribution. This distribution therefore has to be
approximated using the moments and inverse moments of the Wishart

The Wishart distribution can be defined generally on symmetric cones.
We denote by K the orthogonal subgroup of the group of automorphisms of
the cone. We will show how, in this general case, all K-invariant moments for
the Wishart and its inverse can be found using spherical polynomials.

In the particular case of the cone of Hermitian matrices, all moments (not
necessarily K-invariant) can be obtained using the structure of the
symmetric group.