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

TitleCLT for eigenvalue statistics of large dimensional general Fisher matrices and applications
Authors
Issue Date2015
PublisherBernoulli 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?
AbstractSample covariance matrices are widely used in multivariate statistical analysis. The central limit theorems (CLTs) for linear spectral statistics of high-dimensional noncentralized sample covariance matrices have received considerable attention in random matrix theory and have been applied to many high-dimensional statistical problems. However, known population mean vectors are assumed for noncentralized sample covariance matrices, some of which even assume Gaussian-like 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 Gaussian-like moment conditions do not hold but also characterize the nonnegligible differences among the CLTs for the three classes of high-dimensional 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 high-dimensional covariance matrices.
Persistent Identifierhttp://hdl.handle.net/10722/231309
ISSN
2015 Impact Factor: 1.372
2015 SCImago Journal Rankings: 2.120

 

DC FieldValueLanguage
dc.contributor.authorZheng, S-
dc.contributor.authorBai, Z-
dc.contributor.authorYao, JJ-
dc.date.accessioned2016-09-20T05:22:13Z-
dc.date.available2016-09-20T05:22:13Z-
dc.date.issued2015-
dc.identifier.citationBernoulli (Forthcoming)-
dc.identifier.issn1350-7265-
dc.identifier.urihttp://hdl.handle.net/10722/231309-
dc.description.abstractSample covariance matrices are widely used in multivariate statistical analysis. The central limit theorems (CLTs) for linear spectral statistics of high-dimensional noncentralized sample covariance matrices have received considerable attention in random matrix theory and have been applied to many high-dimensional statistical problems. However, known population mean vectors are assumed for noncentralized sample covariance matrices, some of which even assume Gaussian-like 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 Gaussian-like moment conditions do not hold but also characterize the nonnegligible differences among the CLTs for the three classes of high-dimensional 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 high-dimensional covariance matrices.-
dc.languageeng-
dc.publisherBernoulli Society for Mathematical Statistics and Probability. The Journal's web site is located at http://projecteuclid.org/euclid.bj-
dc.relation.ispartofBernoulli-
dc.titleCLT for eigenvalue statistics of large dimensional general Fisher matrices and applications-
dc.typeArticle-
dc.identifier.emailYao, JJ: jeffyao@hku.hk-
dc.identifier.authorityYao, JJ=rp01473-
dc.identifier.hkuros263091-
dc.publisher.placeNetherlands-

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