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

Abstract

Let $\bbZ_{M_1\times N}=\bbT^{\frac{1}{2}}\bbX$ where $(\bbT^{\frac{1}{2}})^2=\bbT$ is a positive definite matrix and $\bbX$ consists of independent random variables with mean zero and variance one. This paper proposes a unified matrix model $$\bold{\bbom}=(\bbZ\bbU_2\bbU_2^T\bbZ^T)^{-1}\bbZ\bbU_1\bbU_1^T\bbZ^T,$$ where $\bbU_1$ and $\bbU_2$ are isometric with dimensions $N\times N_1$ and $N\times (N-N_2)$ respectively such that $\bbU_1^T\bbU_1=\bbI_{N_1}$, $\bbU_2^T\bbU_2=\bbI_{N-N_2}$ and $\bbU_1^T\bbU_2=0$. Moreover, $\bbU_1$ and $\bbU_2$ (random or non-random) are independent of $\bbZ_{M_1\times N}$ and with probability tending to one, $rank(\bbU_1)=N_1$ and $rank(\bbU_2)=N-N_2$. We establish the asymptotic Tracy-Widom distribution for its largest eigenvalue under moment assumptions on $\bbX$ when $N_1,N_2$ and $M_1$ are comparable. By selecting appropriate matrices $\bbU_1$ and $\bbU_2$, 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 $\bold{\bbom}$. %In particular, $\bbom$ can also cover nonzero mean by appropriate matrices $\bbU_1$ and $\bbU_2$. %relax the zero mean value restriction for F matrix in \cite{WY} to allow for any nonzero mean vetors. %thus a direct application of our proposed Tracy-Widom distribution is the independence testing via CCA. Moreover, via appropriate matrices $\bbU_1$ and $\bbU_2$, this matrix $\bold{\bbom}$ can be applied to some multivariate testing problems that cannot be done by the traditional CCA matrix.