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

Hillier, Grant and Kan, Raymond M. (2024) On the expectations of equivariant matrix‐valued functions of Wishart and inverse Wishart matrices. Scandinavian Journal of Statistics, 51 (2), 697-723. (doi:10.1111/sjos.12707).

Record type: Article

Abstract

Many matrix-valued functions of an m x m 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 θ(m), that is, F_k(H W H^T) = H F_k(W)H^T,H εθ(m). It is easy to see that the expectation of such a function is itself homogeneous of degree k in Σ, the covariance matrix, and are also equivariant under the action of θ(m) on Σ. The space of such homogeneous, equivariant, matrix-valued functions is spanned by elements of the type W^rpλ(W), where rε {0,...,k} and, for each r,λ varies over the partitions of k-r, and pλ(W) denotes the power-sum symmetric function indexed by λ. In the analogous case where W is replaced by W^-1, these elements are replaced byW^-rpλ(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 the computation of all such moments.