Estimation of the inverse scatter matrix for a scale mixture of Wishart matrices under Efron–Morris type losses

We consider estimation of the inverse scatter matrix Σ−1 for a scale mixture of Wishart matrices under various Efron–Morris type losses, tr[{Σˆ−1−Σ−1}2Sk] for k=0,1,2..., where S is the sample covariance matrix. We improve on the standard estimators aS+, where S+ denotes the Moore–Penrose inverse of S and a is a positive constant, through an unbiased estimator of the risk difference between the new estimators and aS+. Thus we demonstrate that improvements over the standard estimators under a Wishart distribution can be extended under mixing. We give a unified treatment of the two cases where S is invertible (S+=S−1) and where S is singular.