On the consistent estimators of the population covariance matrix and its reparameterizations

For the high-dimensional covariance estimation problem, when $\lim_{n\to \infty}p/n=c \in (0,1)$ the orthogonally equivariant estimator of the population covariance matrix proposed by Tsai and Tsai (2024b) enjoys some optimal properties. Under some regularity conditions, they showed that their novel estimators of eigenvalues are consistent with the eigenvalues of the population covariance matrix. In this note, first, we show that their novel estimator is a consistent estimator of the population covariance matrix under a high-dimensional asymptotic setup. Moreover, we may show that the novel estimator is the MLE of the population covariance matrix when $c \in (0, 1)$. The novel estimator is incorporated to establish the optimal decomposite $T_{T}^{2}-$test for a high-dimensional statistical hypothesis testing problem and to make the statistical inference for the high-dimensional principal component analysis-related problems without the sparsity assumption. Some remarks when $p >n $, especially for the high-dimensional low-sample size categorical data models $p >> n$, are made in the final section.