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

Many matrix‐valued functions of an m×m Wishart matrix W, Fk(W), say, are homogeneous of degree k in W, and are equivariant under the conjugate action of the orthogonal group 𝒪(m), that is, Fk(HWHT)=HFk(W)HT, H∈𝒪(m). It is easy to see that the expectation of such a function is itself homogeneous of degree k in ∑, the covariance matrix, and are also equivariant under the action of 𝒪(m) on ∑. The space of such homogeneous, equivariant, matrix‐valued functions is spanned by elements of the type Wrpλ(W), where r∈{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 by W−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.