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GENERALIZED LINEAR SPECTRAL STATISTICS OF HIGH-DIMENSIONAL SAMPLE COVARIANCE MATRICES AND ITS APPLICATIONS
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DOI：10.1214/25-AOS2601.png)
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Abstract
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In this paper, we introduce the Generalized Linear Spectral Statistics (GLSS) of a high-dimensional sample covariance matrix Sn, denoted as tr f (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 root N/rank(Bn). 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:
Sample covariance matrix
random matrix theory
eigenspaces
generalized linear spectral statistics
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