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 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.
Funding Statement
G. M. Pan was partially supported by MOE Tier 2 Grant 2018-T2-2-112 and by a MOE Tier 1 Grant RG133/18 at the Nanyang Technological University, Singapore.
S. Zheng (corresponding author) was partially supported by NSFC grant 12071066 and KLAS.
P.-S. Zhong was partially supported by an NSF grant DMS-1462156 and an NIH grant 1R21HG010073.
Acknowledgements
We are grateful to the Editor, the Associate Editor and two referees for their constructive comments, which helped us to improve the manuscript.
Citation
Zhixiang Zhang. Shurong Zheng. Guangming Pan. Ping-Shou Zhong. "Asymptotic independence of spiked eigenvalues and linear spectral statistics for large sample covariance matrices." Ann. Statist. 50 (4) 2205 - 2230, August 2022. https://doi.org/10.1214/22-AOS2183
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