STATISTICAL EIGEN-INFERENCE FROM LARGE WISHART MATRICES

• 成果类型: Article
• 署名作者: Rao, N. Raj; Mingo, James A.; Speicher, Roland; Edelman, Alan
• 署名单位: Massachusetts Institute of Technology (MIT); Massachusetts Institute of Technology (MIT); Queens University - Canada; Massachusetts Institute of Technology (MIT)
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/07-AOS583
• 发表日期: 2008
• 页码: 2850-2885
• 关键词: hypergeometric-functions; laplace approximations; 2nd-order freeness; COVARIANCE-MATRIX; sample; fluctuations; eigenvalues; clt

摘要

We consider settings where the observations are drawn from a zero-mean multivariate (real or complex) normal distribution with the population covariance matrix having eigenvalues of arbitrary multiplicity. We assume that the eigenvectors of the population covariance matrix are unknown and focus on inferential procedures that are based on the sample eigenvalues alone (i.e., eigen-inference). Results found in the literature establish the asymptotic normality of the fluctuation in the trace of powers of the sample covariance matrix. We develop concrete algorithms for analytically computing the limiting quantities and the covariance of the fluctuations. We exploit the asymptotic normality of the trace of powers of the sample covariance matrix to develop eigenvalue-based procedures for testing and estimation. Specifically. we formulate a Simple test of hypotheses for the population eigenvalues and a technique for estimating the population eigenvalues in settings where the cumulative distribution function of the (nonrandom) population eigenvalues has a staircase structure. Monte Carlo simulations are used to demonstrate the superiority of the proposed methodologies over classical techniques and the robustness of the proposed techniques in high-dimensional, (relatively) small sample size settings. The improved performance results from the fact that the proposed inference procedures are global (in a sense that we describe) and exploit global information thereby overcoming the inherent biases that cripple classical inference procedures which are local and rely on local information.