ASYMPTOTIC INDEPENDENCE OF SPIKED EIGENVALUES AND LINEAR SPECTRAL STATISTICS FOR LARGE SAMPLE COVARIANCE MATRICES

• 成果类型: Article
• 署名作者: Zhang, Zhixiang; Zheng, Shurong; Pan, Guangming; Zhong, Ping-Shou
• 署名单位: Nanyang Technological University; Northeast Normal University - China; University of Illinois System; University of Illinois Chicago; University of Illinois Chicago Hospital
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/22-AOS2183
• 发表日期: 2022
• 页码: 2205-2230
• 关键词: 4 moment theorem; High-dimension; clt; eigenstructure; eigenvectors; support; PCA

摘要

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.