An identity for the Wishart distribution with applications

Let Sp-p have a Wishart distribution with unknown matrix [Sigma] and k degrees of freedom. For a matrix T(S) and a scalar h(S), an identity is obtained for E[summation operator]tr[h(S)T[summation operator]-1]. Two applications are given. The first provides product moments and related formulae for the Wishart distribution. Higher moments involving S can be generated recursively. The second application concerns good estimators of [summation operator] and [summation operator]-1. In particular, identities for several risk functions are obtained, and estimators of [summation operator] ([summation operator]-1) are described which dominate aS(bS-1), a <= 1/k (b <= k - p - 1). [3] Ann. Statist. 7 No. 5; (1980) Ann. Statist. 8 used special cases of the identity to find unbiased risk estimators. These are unobtainable in closed form for certain natural loss functions. In this paper, we treat these case as well. The dominance results provide a unified theory for the estimation of [summation operator] and [summation operator]-1.