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

Abstract

Many matrix‐valued functions of an Wishart matrix , , say, are homogeneous of degree in , and are equivariant under the conjugate action of the orthogonal group , that is, , . It is easy to see that the expectation of such a function is itself homogeneous of degree in , the covariance matrix, and are also equivariant under the action of on . The space of such homogeneous, equivariant, matrix‐valued functions is spanned by elements of the type , where and, for each , varies over the partitions of , and denotes the power‐sum symmetric function indexed by . In the analogous case where is replaced by , these elements are replaced by . 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.