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Article: CLT for eigenvalue statistics of large dimensional general Fisher matrices and applications
Title  CLT for eigenvalue statistics of large dimensional general Fisher matrices and applications 

Authors  
Issue Date  2015 
Publisher  Bernoulli Society for Mathematical Statistics and Probability. The Journal's web site is located at http://projecteuclid.org/euclid.bj 
Citation  Bernoulli (Forthcoming) How to Cite? 
Abstract  Sample covariance matrices are widely used in multivariate statistical analysis. The central limit theorems (CLTs) for linear spectral statistics of highdimensional noncentralized sample covariance matrices have received considerable attention in random matrix theory and have been applied to many highdimensional statistical problems. However, known population mean vectors are assumed for noncentralized sample covariance matrices, some of which even assume Gaussianlike moment conditions. In fact, there are still another two most frequently used sample covariance matrices: the ME (moment estimator, constructed by subtracting the sample mean vector from each sample vector) and the unbiased sample covariance matrix (by changing the denominator nn as N=n−1N=n−1 in the ME) without depending on unknown population mean vectors. In this paper, we not only establish the new CLTs for noncentralized sample covariance matrices when the Gaussianlike moment conditions do not hold but also characterize the nonnegligible differences among the CLTs for the three classes of highdimensional sample covariance matrices by establishing a substitution principle: by substituting the adjusted sample size N=n−1N=n−1 for the actual sample size nn in the centering term of the new CLTs, we obtain the CLT of the unbiased sample covariance matrices. Moreover, it is found that the difference between the CLTs for the ME and unbiased sample covariance matrix is nonnegligible in the centering term although the only difference between two sample covariance matrices is a normalization by nn and n−1n−1, respectively. The new results are applied to two testing problems for highdimensional covariance matrices. 
Persistent Identifier  http://hdl.handle.net/10722/231309 
ISSN  2015 Impact Factor: 1.372 2015 SCImago Journal Rankings: 2.120 
DC Field  Value  Language 

dc.contributor.author  Zheng, S   
dc.contributor.author  Bai, Z   
dc.contributor.author  Yao, JJ   
dc.date.accessioned  20160920T05:22:13Z   
dc.date.available  20160920T05:22:13Z   
dc.date.issued  2015   
dc.identifier.citation  Bernoulli (Forthcoming)   
dc.identifier.issn  13507265   
dc.identifier.uri  http://hdl.handle.net/10722/231309   
dc.description.abstract  Sample covariance matrices are widely used in multivariate statistical analysis. The central limit theorems (CLTs) for linear spectral statistics of highdimensional noncentralized sample covariance matrices have received considerable attention in random matrix theory and have been applied to many highdimensional statistical problems. However, known population mean vectors are assumed for noncentralized sample covariance matrices, some of which even assume Gaussianlike moment conditions. In fact, there are still another two most frequently used sample covariance matrices: the ME (moment estimator, constructed by subtracting the sample mean vector from each sample vector) and the unbiased sample covariance matrix (by changing the denominator nn as N=n−1N=n−1 in the ME) without depending on unknown population mean vectors. In this paper, we not only establish the new CLTs for noncentralized sample covariance matrices when the Gaussianlike moment conditions do not hold but also characterize the nonnegligible differences among the CLTs for the three classes of highdimensional sample covariance matrices by establishing a substitution principle: by substituting the adjusted sample size N=n−1N=n−1 for the actual sample size nn in the centering term of the new CLTs, we obtain the CLT of the unbiased sample covariance matrices. Moreover, it is found that the difference between the CLTs for the ME and unbiased sample covariance matrix is nonnegligible in the centering term although the only difference between two sample covariance matrices is a normalization by nn and n−1n−1, respectively. The new results are applied to two testing problems for highdimensional covariance matrices.   
dc.language  eng   
dc.publisher  Bernoulli Society for Mathematical Statistics and Probability. The Journal's web site is located at http://projecteuclid.org/euclid.bj   
dc.relation.ispartof  Bernoulli   
dc.title  CLT for eigenvalue statistics of large dimensional general Fisher matrices and applications   
dc.type  Article   
dc.identifier.email  Yao, JJ: jeffyao@hku.hk   
dc.identifier.authority  Yao, JJ=rp01473   
dc.identifier.hkuros  263091   
dc.publisher.place  Netherlands   