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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  2017 
Publisher  Institute of Mathematical Statistics. 
Citation  The Annals of Statistics, 2017, v. 45 n. 1, p. 415460 How to Cite? 
Abstract  Consider two ppvariate populations, not necessarily Gaussian, with covariance matrices Σ1Σ1 and Σ2Σ2, respectively. Let S1S1 and S2S2 be the corresponding sample covariance matrices with degrees of freedom mm and nn. When the difference ΔΔ between Σ1Σ1 and Σ2Σ2 is of small rank compared to p,mp,m and nn, the Fisher matrix S:=S−12S1S:=S2−1S1 is called a spiked Fisher matrix. When p,mp,m and nn 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 ΔΔ (spikes) are above (or under) a critical value, the associated extreme eigenvalues of SS 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 SS. The limiting distributions are found to be Gaussian if and only if the corresponding population spike eigenvalues in ΔΔ are 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  2017   
dc.identifier.citation  The Annals of Statistics, 2017, v. 45 n. 1, p. 415460   
dc.identifier.issn  00905364   
dc.identifier.uri  http://hdl.handle.net/10722/231315   
dc.description.abstract  Consider two ppvariate populations, not necessarily Gaussian, with covariance matrices Σ1Σ1 and Σ2Σ2, respectively. Let S1S1 and S2S2 be the corresponding sample covariance matrices with degrees of freedom mm and nn. When the difference ΔΔ between Σ1Σ1 and Σ2Σ2 is of small rank compared to p,mp,m and nn, the Fisher matrix S:=S−12S1S:=S2−1S1 is called a spiked Fisher matrix. When p,mp,m and nn 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 ΔΔ (spikes) are above (or under) a critical value, the associated extreme eigenvalues of SS 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 SS. The limiting distributions are found to be Gaussian if and only if the corresponding population spike eigenvalues in ΔΔ are 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.publisher  Institute of Mathematical Statistics.   
dc.relation.ispartof  The Annals of Statistics   
dc.rights  This work is licensed under a Creative Commons AttributionNonCommercialNoDerivatives 4.0 International License.   
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.description.nature  published_or_final_version   
dc.identifier.doi  10.1214/16AOS1463   
dc.identifier.hkuros  263179   
dc.identifier.volume  45   
dc.identifier.issue  1   
dc.identifier.spage  415   
dc.identifier.epage  460   
dc.publisher.place  United States   