Suppose that n independent observations are drawn from a multivariate normal distribution Np (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 - Ip)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.].
- Best lower triangular equivariant minimax estimator
- Bivariate normal distribution
- Precision matrix
- The squared loss