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 N-p(mu, Sigma) with both mean vector p and covariance matrix E unknown. We consider the problem of estimating the precision matrix Sigma(-1) under the squared loss L((Sigma) over cap (-1), Sigma(-1)) = tr((Sigma) over cap (-1)Sigma - I-p)(2). 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.] (c) 2007 Elsevier B.V. All rights reserved.