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), aWishart and inverted moments estimation of covariance matrix and its inverse Stoke's theorem general identities for the risk function