MULTIVARIATE ANALYSIS AND JACOBI ENSEMBLES: LARGEST EIGENVALUE, TRACY-WIDOM LIMITS AND RATES OF CONVERGENCE

• 成果类型: Article
• 署名作者: Johnstone, Iain M.
• 署名单位: Stanford University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/08-AOS605
• 发表日期: 2008
• 页码: 2638-2716
• 关键词: orthogonal polynomials; spacing distributions; asymptotics; spectrum; UNIVERSALITY; unitary; edge

摘要

Let A and B be independent, central Wishart matrices in p variables with common covariance and having in and n degrees of freedom, respectively. The distribution of the largest eigenvalue of (A + B)(-1) B has numerous applications in multivariate statistics, but is difficult to calculate exactly. Suppose that in and n grow in proportion to p. We show that after centering and scaling, the distribution is approximated to second-order, O(p(-2/3)), by the Tracy-Widom law. The results are obtained for both complex and then real-valued data by using methods of random matrix theory to study the largest eigenvalue of the Jacobi unitary and orthogonal ensembles. Asymptotic approximations of Jacobi polynomials near the largest zero play a central role.