Generalized linear spectral statistics of high-dimensional sample covariance matrices and its applications
Yanlin Hu, Qing Yang, Xiao Han
The Annals of Statistics · 2026 · Volume 54 · Issue 2
Abstract
In this paper, we introduce the Generalized Linear Spectral Statistics (GLSS) of a high-dimensional sample covariance matrix Sn, denoted as trf(Sn)Bn, which effectively captures distinct spectral properties of Sn by incorporating an ancillary matrix Bn and a test function f. The joint asymptotic normality of GLSS associated with different test functions is established under mild assumptions on Bn and the underlying distribution, when the dimension n and sample size N are comparable. The convergence rate of GLSS is determined by N/rank(B n). Subsequently, we propose a novel functional projection approach based on GLSS for hypothesis testing on eigenspaces of “population-spiked” covariance matrices, showcasing a universality phenomenon in the magnitude of the spikes. The theoretical accuracy of our results established for GLSS and the advantages of the newly suggested testing procedure are demonstrated through various numerical studies.
Keywords
60B20; 60F05; 62H10; 62H15; eigenspaces; generalized linear spectral statistics; Random matrix theory; Sample covariance matrix;
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