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#### Article: EXTREME EIGENVALUES OF LARGE-DIMENSIONAL SPIKED FISHER MATRICES WITH APPLICATION

Title EXTREME EIGENVALUES OF LARGE-DIMENSIONAL SPIKED FISHER MATRICES WITH APPLICATION Wang, QYao, JJ 2016 The Annals of Statistics (Forthcoming) How to Cite? 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 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. http://hdl.handle.net/10722/231315 0090-53642015 Impact Factor: 2.782015 SCImago Journal Rankings: 6.653

DC FieldValueLanguage
dc.contributor.authorWang, Q-
dc.contributor.authorYao, JJ-
dc.date.accessioned2016-09-20T05:22:16Z-
dc.date.available2016-09-20T05:22:16Z-
dc.date.issued2016-
dc.identifier.citationThe Annals of Statistics (Forthcoming)-
dc.identifier.issn0090-5364-
dc.identifier.urihttp://hdl.handle.net/10722/231315-
dc.description.abstractConsider 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 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.-
dc.languageeng-
dc.relation.ispartofThe Annals of Statistics-
dc.titleEXTREME EIGENVALUES OF LARGE-DIMENSIONAL SPIKED FISHER MATRICES WITH APPLICATION-
dc.typeArticle-
dc.identifier.emailYao, JJ: jeffyao@hku.hk-
dc.identifier.authorityYao, JJ=rp01473-
dc.identifier.hkuros263179-