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

TL;DR

This paper demonstrates that a recently proposed orthogonally equivariant estimator of the population covariance matrix is consistent and can be viewed as the maximum likelihood estimator in high-dimensional settings, improving inference in PCA and hypothesis testing.

Contribution

It proves the consistency and MLE property of a novel high-dimensional covariance estimator and applies it to develop optimal tests and inference methods without sparsity assumptions.

Findings

Estimator is consistent under high-dimensional asymptotics.

Estimator is the MLE when the ratio c is in (0,1).

Application to high-dimensional PCA and hypothesis testing.

Abstract

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.