  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.
