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.

翻译：本文在维度$p$较大的情况下，深入探究了Mar$\check{c}$enko-Pastur方程的内在机理，得到了一个显式的等式关系，并利用该等式构建了一类新的总体协方差矩阵的正交等变估计量。在一定的正则性条件下，证明了所提出的总体特征值估计量是总体协方差矩阵特征值的一致估计量。同时证明了在归一化Stein损失函数下，所提出的估计量是总体协方差矩阵的最优正交等变估计量。