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