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November 2018 A unified matrix model including both CCA and F matrices in multivariate analysis: The largest eigenvalue and its applications
Xiao Han, Guangming Pan, Qing Yang
Bernoulli 24(4B): 3447-3468 (November 2018). DOI: 10.3150/17-BEJ965

Abstract

Let $\mathbf{Z}_{M_{1}\times N}=\mathbf{T}^{\frac{1}{2}}\mathbf{X}$ where $(\mathbf{T}^{\frac{1}{2}})^{2}=\mathbf{T}$ is a positive definite matrix and $\mathbf{X}$ consists of independent random variables with mean zero and variance one. This paper proposes a unified matrix model \[\mathbf{\Omega}=(\mathbf{Z}\mathbf{U}_{2}\mathbf{U}_{2}^{T}\mathbf{Z}^{T})^{-1}\mathbf{Z}\mathbf{U}_{1}\mathbf{U}_{1}^{T}\mathbf{Z}^{T},\] where $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$ are isometric with dimensions $N\times N_{1}$ and $N\times(N-N_{2})$ respectively such that $\mathbf{U}_{1}^{T}\mathbf{U}_{1}=\mathbf{I}_{N_{1}}$, $\mathbf{U}_{2}^{T}\mathbf{U}_{2}=\mathbf{I}_{N-N_{2}}$ and $\mathbf{U}_{1}^{T}\mathbf{U}_{2}=0$. Moreover, $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$ (random or non-random) are independent of $\mathbf{Z}_{M_{1}\times N}$ and with probability tending to one, $\operatorname{rank}(\mathbf{U}_{1})=N_{1}$ and $\operatorname{rank}(\mathbf{U}_{2})=N-N_{2}$. We establish the asymptotic Tracy–Widom distribution for its largest eigenvalue under moment assumptions on $\mathbf{X}$ 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 $\mathbf{\Omega}$ by selecting appropriate matrices $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$. Moreover, via appropriate matrices $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$, this matrix $\mathbf{\Omega}$ 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 $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$ are chosen to be two nonrandom matrices. The other one is in the multivariate linear model for testing the unknown parameter matrix, where $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$ are random. For each application, theoretical results are developed and various numerical studies are conducted to investigate the empirical performance.

Citation

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Xiao Han. Guangming Pan. Qing Yang. "A unified matrix model including both CCA and F matrices in multivariate analysis: The largest eigenvalue and its applications." Bernoulli 24 (4B) 3447 - 3468, November 2018. https://doi.org/10.3150/17-BEJ965

Information

Received: 1 January 2017; Revised: 1 June 2017; Published: November 2018
First available in Project Euclid: 18 April 2018

zbMATH: 06869881
MathSciNet: MR3788178
Digital Object Identifier: 10.3150/17-BEJ965

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

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Vol.24 • No. 4B • November 2018
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