Substitution principle for CLT of linear spectral statistics of high-dimensional sample covariance matrices with applications to hypothesis testing

TL;DR

This paper establishes new central limit theorems for high-dimensional non-centered sample covariance matrices, introduces a substitution principle to relate different estimators, and applies these results to hypothesis testing.

Contribution

It provides a novel substitution principle for CLTs of high-dimensional covariance matrices, removing Gaussian-like moment restrictions and characterizing differences among estimators.

Findings

New CLTs for non-centered covariance matrices without Gaussian assumptions

Substitution principle linking CLTs of different covariance estimators

Non-negligible difference in CLT centering terms between MLE and unbiased estimators

Abstract

Sample covariance matrices are widely used in multivariate statistical analysis. The central limit theorems (CLT's) for linear spectral statistics of high-dimensional non-centered sample covariance matrices have received considerable attention in random matrix theory and have been applied to many high-dimensional statistical problems. However, known population mean vectors are assumed for non-centered sample covariance matrices, some of which even assume Gaussian-like moment conditions. In fact, there are still another two most frequently used sample covariance matrices: the MLE (by subtracting the sample mean vector from each sample vector) and the unbiased sample covariance matrix (by changing the denominator $n$ as $N=n-1$ in the MLE) without depending on unknown population mean vectors. In this paper, we not only establish new CLT's for non-centered sample covariance matrices without Gaussian-like moment conditions but also characterize the non-negligible differences among the CLT's for the three classes of high-dimensional sample covariance matrices by establishing a {\em substitution principle}: substitute the {\em adjusted} sample size $N=n-1$ for the actual sample size $n$ in the major centering term of the new CLT's so as to obtain the CLT of the unbiased sample covariance matrices. Moreover, it is found that the difference between the CLT's for the MLE and unbiased sample covariance matrix is non-negligible in the major centering term although the two sample covariance matrices only have differences $n$ and $n-1$ on the dominator. The new results are applied to two testing problems for high-dimensional data.