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

Abstract

Let ZM1×N = T 12 X where (T 12 )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 = ZU2UT2 ZT−1ZU1UT1 ZT , where U1 and U2 are isometric with dimensions N × N1 and N × (N − N2) respectively such that UT1 U1 = IN1 , UT2 U2 = IN−N2 and UT1 U2 = 0. Moreover, U1 and U2 (random or non-random) are independent of ZM1×N and with probability tending to one, rank(U1) = N1 and rank(U2) = N − N2. We establish the asymptotic Tracy–Widom distribution for its largest eigenvalue under moment assumptions on X when N1, N2 and M1 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 by selecting appropriate matrices U1 and U2. Moreover, via appropriate matrices U1 and U2, this matrix 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 U1 and U2 are chosen to be two nonrandom matrices. The other one is in the multivariate linear model for testing the unknown parameter matrix, where U1 and U2 are random. For each application, theoretical results are developed and various numerical studies are conducted to investigate the empirical performance.