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Article: EXTREME EIGENVALUES OF LARGEDIMENSIONAL SPIKED FISHER MATRICES WITH APPLICATION
Title  EXTREME EIGENVALUES OF LARGEDIMENSIONAL SPIKED FISHER MATRICES WITH APPLICATION 

Authors  
Issue Date  2016 
Citation  The Annals of Statistics (Forthcoming) How to Cite? 
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 {\em 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 $\Delta$ ({\em 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 {\em simple}. Two applications are introduced. The first application uses the largest eigenvalue of the Fisher matrix to test the equality between two highdimensional 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. 
Persistent Identifier  http://hdl.handle.net/10722/231315 
ISSN  2015 Impact Factor: 2.78 2015 SCImago Journal Rankings: 6.653 
DC Field  Value  Language 

dc.contributor.author  Wang, Q   
dc.contributor.author  Yao, JJ   
dc.date.accessioned  20160920T05:22:16Z   
dc.date.available  20160920T05:22:16Z   
dc.date.issued  2016   
dc.identifier.citation  The Annals of Statistics (Forthcoming)   
dc.identifier.issn  00905364   
dc.identifier.uri  http://hdl.handle.net/10722/231315   
dc.description.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 {\em 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 $\Delta$ ({\em 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 {\em simple}. Two applications are introduced. The first application uses the largest eigenvalue of the Fisher matrix to test the equality between two highdimensional 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.   
dc.language  eng   
dc.relation.ispartof  The Annals of Statistics   
dc.title  EXTREME EIGENVALUES OF LARGEDIMENSIONAL SPIKED FISHER MATRICES WITH APPLICATION   
dc.type  Article   
dc.identifier.email  Yao, JJ: jeffyao@hku.hk   
dc.identifier.authority  Yao, JJ=rp01473   
dc.identifier.hkuros  263179   