## 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

distribution.

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.