Improved minimax estimation of the bivariate normal precision matrix under the squared loss

Suppose that n independent observations are drawn from a multivariate normal distribution Np([mu],[Sigma]) with both mean vector [mu] and covariance matrix [Sigma] unknown. We consider the problem of estimating the precision matrix [Sigma]-1 under the squared loss . It is well known that the best lower triangular equivariant estimator of [Sigma]-1 is minimax. In this paper, by using the information in the sample mean on [Sigma]-1, we construct a new class of improved estimators over the best lower triangular equivariant minimax estimator of [Sigma]-1 for p=2. Our improved estimators are in the class of lower-triangular scale equivariant estimators and the method used is similar to that of Stein [1964. Inadmissibility of the usual estimator for the variance of a normal distribution with unknown mean. Ann. Inst. Statist. Math. 16, 155-160.]