A0838
Title: Statistical inference for principal components of spiked covariance matrices
Authors: Jingming Wang - Harvard University (United States) [presenting]
Zhigang Bao - Hong Kong University of Science and Technology (Hong Kong)
Xiucai Ding - UC Davis (United States)
Ke Wang - Hong Kong University of Science and Technology (Hong Kong)
Abstract: In random matrix theory, one of the central topics is the limiting behavior of eigenvalues and eigenvectors of random matrices under fixed-rank perturbations. A famous model raised by Johnstone is the so-called spiked covariance matrix model. It is a sample covariance matrix whose population has all its eigenvalues equal to one except for a few top eigenvalues (spikes). From the principal component analysis (PCA) point of view, the main task is to study the limiting behavior of the top eigenvalues and eigenvectors of the spiked sample covariance matrix. The high dimensional setting is considered; namely, both the sample size n and the dimension p are large. The limiting distribution of the eigenvectors associated with the largest eigenvalues is first identified for the sample covariance matrix in the supercritical regime. Second, the joint distribution is derived from the extreme eigenvalues and the associated eigenvectors. Third, based on these results, accurate and powerful statistics are proposed, and their asymptotic distributions are derived in order to conduct hypothesis testing on the principal components. Numerical simulations also confirm the accuracy and power of the proposed statistics and illustrate significantly better performance compared to the existing methods in the literature.
