The Asymptotic Properties of the Extreme Eigenvectors of High-dimensional Generalized Spiked Covariance Model

TL;DR

This paper studies the asymptotic behavior of the extreme eigenvectors in high-dimensional spiked covariance models, removing previous restrictions and providing new theoretical insights and a hypothesis testing statistic.

Contribution

It extends the analysis of eigenvector asymptotics to more general models without block diagonal assumptions or bounded eigenvalues, using advanced random matrix theory techniques.

Findings

Derived convergence and limiting distributions for eigenvector projections

Proposed a new hypothesis testing statistic for eigenspaces

Demonstrated robustness of methods even with large eigenvalue differences

Abstract

In this paper, we investigate the asymptotic behaviors of the extreme eigenvectors in a general spiked covariance matrix, where the dimension and sample size increase proportionally. We eliminate the restrictive assumption of the block diagonal structure in the population covariance matrix. Moreover, there is no requirement for the spiked eigenvalues and the 4th moment to be bounded. Specifically, we apply random matrix theory to derive the convergence and limiting distributions of certain projections of the extreme eigenvectors in a large sample covariance matrix within a generalized spiked population model. Furthermore, our techniques are robust and effective, even when spiked eigenvalues differ significantly in magnitude from nonspiked ones. Finally, we propose a powerful statistic for hypothesis testing for the eigenspaces of covariance matrices.