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February 2017 Extreme eigenvalues of large-dimensional spiked Fisher matrices with application
Qinwen Wang, Jianfeng Yao
Ann. Statist. 45(1): 415-460 (February 2017). DOI: 10.1214/16-AOS1463

Abstract

Consider two $p$-variate populations, not necessarily Gaussian, with covariance matrices $\Sigma_{1}$ and $\Sigma_{2}$, respectively. Let $S_{1}$ and $S_{2}$ be the corresponding sample covariance matrices with degrees of freedom $m$ and $n$. When the difference $\Delta$ between $\Sigma_{1}$ and $\Sigma_{2}$ is of small rank compared to $p,m$ and $n$, the Fisher matrix $S:=S_{2}^{-1}S_{1}$ is called a spiked Fisher matrix. When $p,m$ and $n$ grow to infinity proportionally, we establish a phase transition for the extreme eigenvalues of the Fisher matrix: a displacement formula showing that when the eigenvalues of $\Delta$ (spikes) are above (or under) a critical value, the associated extreme eigenvalues of $S$ will converge to some point outside the support of the global limit (LSD) of other eigenvalues (become outliers); otherwise, they will converge to the edge points of the LSD. Furthermore, we derive central limit theorems for those outlier eigenvalues of $S$. The limiting distributions are found to be Gaussian if and only if the corresponding population spike eigenvalues in $\Delta$ are simple. Two applications are introduced. The first application uses the largest eigenvalue of the Fisher matrix to test the equality between two high-dimensional covariance matrices, and explicit power function is found under the spiked alternative. The second application is in the field of signal detection, where an estimator for the number of signals is proposed while the covariance structure of the noise is arbitrary.

Citation

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Qinwen Wang. Jianfeng Yao. "Extreme eigenvalues of large-dimensional spiked Fisher matrices with application." Ann. Statist. 45 (1) 415 - 460, February 2017. https://doi.org/10.1214/16-AOS1463

Information

Received: 1 April 2015; Revised: 1 March 2016; Published: February 2017
First available in Project Euclid: 21 February 2017

zbMATH: 1360.62283
MathSciNet: MR3611497
Digital Object Identifier: 10.1214/16-AOS1463

Subjects:
Primary: 62H12
Secondary: 60F05

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 1 • February 2017
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