On the expectations of equivariant matrix-valued functions of Wishart and inverse Wishart matrices

Many matrix-valued functions of an mxm Wishart matrix W, F_k(W), say, are homogeneous of degree k in W, and are equivariant under the conjugate action of the orthogonal group O(m), i.e., F_k(HWH')=HF_k(W)H', H \in O(m). It is easy to see that the expectation of such a function is itself homogeneous of degree k in \Sigma, the covariance matrix, and are also equivariant under the action of O(m) on \Sigma. The space of such homogeneous, equivariant, matrix-valued functions is spanned by elements of the type W^r*p_{\lambda}(W), where r \in {0,...,k} and, for each r, \lambda varies over the partitions of k-r. Here, p_{\lambda}(W) denotes the power-sum symmetric function indexed by \lambda. In the analogous case where W is replaced by W^{-1}, these elements are replaced by W^{-r}*p_{\lambda}(W^{-1}). In this paper we derive recurrence relations and analytical expressions for the expectations of such functions. Our results provide highly efficient methods for analysing the properties of, and the computation of, all such moments, even those of very high order k. We thus provide a complete toolbox for analyzing the properties of any matrix-valued function in this class