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

Abstract

We consider estimation of the inverse scatter matrix Σ −1 for a scale mixture of Wishart matrices under various Efron-Morris type losses, tr[{Σ −1 − Σ −1 } 2 S k ] for k = 0, 1, 2..., where S is the sample covariance matrix. We improve on the standard estimators a S + , 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 a S + . Thus we demontrate that improvements over the standard estimators under a Wishart distribution can be extended under mixing. We give a unified treatement of the two cases where S is invertible (S + = S −1 ) and where S is singular.