A unified matrix model including both CCA and F matrices in multivariate analysis: The largest eigenvalue and its applications

作     者：Han, Xiao Pan, Guangming Yang, Qing 

作者机构：Nanyang Technol Univ Sch Phys & Math Sci Singapore 637371 Singapore 

出 版 物：《BERNOULLI》 (Bernoulli)

年 卷 期：2018年第24卷第4B期

页      面：3447-3468页

核心收录：

学科分类：02[经济学] 0202[经济学-应用经济学] 020208[经济学-统计学] 07[理学] 0714[理学-统计学（可授理学、经济学学位）] 070103[理学-概率论与数理统计] 0701[理学-数学] 

基　　金：MOE Tier 2 Grant [2014-T2-2-060] MOE Tier 1 Grant at the Nanyang Technological University, Singapore [RG25/14] 

主　　题：canonical correlation analysis F matrix largest eigenvalue MANOVA multivariate linear model random matrix theory Tracy-Widom distribution 

摘      要：Let Z(M1xN) = T-1/2 X where (T-1/2)(2) = T is a positive definite matrix and X consists of independent random variables with mean zero and variance one. This paper proposes a unified matrix model Omega = (ZU(2)U(2)(T)Z(T))(-1)ZU(1)U(1)(T)Z(T), where U-1 and U-2 are isometric with dimensions N x N-1 and N x (N - N-2) respectively such that (U1U1)-U-T = IN1, (U2U2)-U-T = I-N - N-2 and (U1U2)-U-T = 0. Moreover, U-1 and U-2 (random or non-random) are independent of Z(M1xN) and with probability tending to one, rank(U-1) = N1 and rank(U-2) = N - N-2. We establish the asymptotic Tracy-Widom distribution for its largest eigenvalue under moment assumptions on X when N-1, N-2 and M-1 are comparable. The asymptotic distributions of the maximum eigenvalues of the matrices used in Canonical Correlation Analysis (CCA) and of F matrices (including centered and non-centered versions) can be both obtained from that of Omega by selecting appropriate matrices U-1 and U-2. Moreover, via appropriate matrices U-1 and U-2, this matrix Omega can be applied to some multivariate testing problems that cannot be done by both types of matrices. To see this, we explore two more applications. One is in the MANOVA approach for testing the equivalence of several high-dimensional mean vectors, where U-1 and U-2 are chosen to be two nonrandom matrices. The other one is in the multivariate linear model for testing the unknown parameter matrix, where U-1 and U-2 are random. For each application, theoretical results are developed and various numerical studies are conducted to investigate the empirical performance.