For Selliah's [5] and Stein's [3] loss functions, Sharmaand Krishnamoorthy [6] have considered orthogonal equivariant estimators $\Psi$. In this paper, we obtain asymptotic risk expressions for $\Psi$ and make a numerical comparison with those for Haff's [2] and Sugiura-Fujimoto [7] estimators. It is observed that $\Psi$ is uniformly better than Haff's estimator and better than Sugiura-Fujimoto estimator except in a small region of the parameter space.

In the problem of estimating the covariance matrix of a multivariate normal population James and Stein (1961) obtained a minimax estimator by considering the best invariant estimator with respect to the triangular group. In this paper we propose an orthogonally invariant estimator obtained by averaging the minimax estimator with respect to the invariant measure on the orthogonal group. Explicit forms of the proposed estimator are given for dimensions 2 and 3. Risk is evaluated for various population covariance matrices and it shows a substantial improvement over the minimax estimator for a wide range of population covariance matrices.