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：本文中，当维度$p$较大时，通过深入分析Mar$\check{c}$enko-Pastur方程以获取显式等式关系，并利用该等式建立了一类新的总体协方差矩阵的正交等变估计量。在一定的正则条件下，所提出的总体特征值新估计量被证明对总体协方差矩阵的特征值具有一致性。同时证明，在标准化 Stein 损失函数下，所提估计量是总体协方差矩阵的最优正交等变估计量。