On the orthogonally equivariant estimators of a covariance matrix

In this note, when the dimension $p$ is large we look into the insight of the Mar$\check{c}$enko-Pastur equation to get an explicit equality relationship, and use the obtained equality to establish a new kind of orthogonally equivariant estimator of the population covariance matrix. Under some regularity conditions, the proposed novel estimators of the population eigenvalues are shown to be consistent for the eigenvalues of population covariance matrix. It is also shown that the proposed estimator is the best orthogonally equivariant estimator of population covariance matrix under the normalized Stein loss function.