Asymptotic independence of spiked eigenvalues and linear spectral statistics for large sample covariance matrices

TL;DR

This paper demonstrates the asymptotic independence between leading sample spiked eigenvalues and linear spectral statistics in high-dimensional covariance models, and introduces a new test for covariance matrix equality.

Contribution

It establishes the asymptotic independence without the block diagonal assumption and proposes consistent estimators for the $L_4$ norm of spiked eigenvectors.

Findings

Asymptotic independence of eigenvalues and spectral statistics

Central limit theorem for spiked eigenvalues without block diagonal assumption

A new powerful test for equality of covariance matrices

Abstract

We consider general high-dimensional spiked sample covariance models and show that their leading sample spiked eigenvalues and their linear spectral statistics are asymptotically independent when the sample size and dimension are proportional to each other. As a byproduct, we also establish the central limit theorem of the leading sample spiked eigenvalues by removing the block diagonal assumption on the population covariance matrix, which is commonly needed in the literature. Moreover, we propose consistent estimators of the $L_4$ norm of the spiked population eigenvectors. Based on these results, we develop a new statistic to test the equality of two spiked population covariance matrices. Numerical studies show that the new test procedure is more powerful than some existing methods.