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(μ,Σ)Np(μ,Σ) with both mean vector μμ and covariance matrix ΣΣ unknown. We consider the problem of estimating the precision matrix Σ-1Σ-1 under the squared loss L(Σ^-1,Σ-1)=tr(Σ^-1Σ-Ip)2. It is well known that the best lower triangular equivariant estimator of Σ-1Σ-1 is minimax. In this paper, by using the information in the sample mean on Σ-1Σ-1, we construct a new class of improved estimators over the best lower triangular equivariant minimax estimator of Σ-1Σ-1 for p=2p=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.]