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Article: On sample eigenvalues in a generalized spiked population model

TitleOn sample eigenvalues in a generalized spiked population model
Authors
KeywordsCentral limit theorems
Extreme eigenvalues
Largest eigenvalue
Primary
Sample covariance matrices
Secondary
Spiked population model
Issue Date2012
PublisherAcademic Press. The Journal's web site is located at http://www.elsevier.com/locate/jmva
Citation
Journal of Multivariate Analysis, 2012, v. 106, p. 167-177 How to Cite?
AbstractIn the spiked population model introduced by Johnstone (2001) [11], the population covariance matrix has all its eigenvalues equal to unit except for a few fixed eigenvalues (spikes). The question is to quantify the effect of the perturbation caused by the spike eigenvalues. Baik and Silverstein (2006) [5] establishes the almost sure limits of the extreme sample eigenvalues associated to the spike eigenvalues when the population and the sample sizes become large. In a recent work Bai and Yao (2008) [4], we have provided the limiting distributions for these extreme sample eigenvalues. In this paper, we extend this theory to a generalized spiked population model where the base population covariance matrix is arbitrary, instead of the identity matrix as in Johnstone's case. As the limiting spectral distribution is arbitrary here, new mathematical tools, different from those in Baik and Silverstein (2006) [5], are introduced for establishing the almost sure convergence of the sample eigenvalues generated by the spikes. © 2011 Elsevier Inc.
Persistent Identifierhttp://hdl.handle.net/10722/143793
ISSN
2015 Impact Factor: 0.857
2015 SCImago Journal Rankings: 1.458
ISI Accession Number ID
Funding AgencyGrant Number
Chinese NSF1171057
University of Hong Kong2010
Funding Information:

We are grateful to the referees for their very careful reading. Their comments have led to significant improvements of the proofs of Theorems 4.1 and 4.2 and a more complete biography on considered subjects. The first author's research is partly supported by a Chinese NSF grant (1171057). The second author's research is supported by a Start-up Research Fund (2010) from The University of Hong Kong.

References

 

DC FieldValueLanguage
dc.contributor.authorBai, Zen_HK
dc.contributor.authorYao, Jen_HK
dc.date.accessioned2011-12-21T08:55:57Z-
dc.date.available2011-12-21T08:55:57Z-
dc.date.issued2012en_HK
dc.identifier.citationJournal of Multivariate Analysis, 2012, v. 106, p. 167-177en_HK
dc.identifier.issn0047-259Xen_HK
dc.identifier.urihttp://hdl.handle.net/10722/143793-
dc.description.abstractIn the spiked population model introduced by Johnstone (2001) [11], the population covariance matrix has all its eigenvalues equal to unit except for a few fixed eigenvalues (spikes). The question is to quantify the effect of the perturbation caused by the spike eigenvalues. Baik and Silverstein (2006) [5] establishes the almost sure limits of the extreme sample eigenvalues associated to the spike eigenvalues when the population and the sample sizes become large. In a recent work Bai and Yao (2008) [4], we have provided the limiting distributions for these extreme sample eigenvalues. In this paper, we extend this theory to a generalized spiked population model where the base population covariance matrix is arbitrary, instead of the identity matrix as in Johnstone's case. As the limiting spectral distribution is arbitrary here, new mathematical tools, different from those in Baik and Silverstein (2006) [5], are introduced for establishing the almost sure convergence of the sample eigenvalues generated by the spikes. © 2011 Elsevier Inc.en_HK
dc.languageengen_US
dc.publisherAcademic Press. The Journal's web site is located at http://www.elsevier.com/locate/jmvaen_HK
dc.relation.ispartofJournal of Multivariate Analysisen_HK
dc.rightsNOTICE: this is the author’s version of a work that was accepted for publication in Journal of Multivariate Analysis. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Multivariate Analysis, 2012, v. 106, p. 167-177. DOI: 10.1016/j.jmva.2011.10.009-
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.subjectCentral limit theoremsen_HK
dc.subjectExtreme eigenvaluesen_HK
dc.subjectLargest eigenvalueen_HK
dc.subjectPrimaryen_HK
dc.subjectSample covariance matricesen_HK
dc.subjectSecondaryen_HK
dc.subjectSpiked population modelen_HK
dc.titleOn sample eigenvalues in a generalized spiked population modelen_HK
dc.typeArticleen_HK
dc.identifier.emailYao, J: jeffyao@hku.hken_HK
dc.identifier.authorityYao, J=rp01473en_HK
dc.description.naturepostprint-
dc.identifier.doi10.1016/j.jmva.2011.10.009en_HK
dc.identifier.scopuseid_2-s2.0-84855813409en_HK
dc.identifier.hkuros198156en_US
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-84855813409&selection=ref&src=s&origin=recordpageen_HK
dc.identifier.volume106en_HK
dc.identifier.spage167en_HK
dc.identifier.epage177en_HK
dc.identifier.isiWOS:000300913400011-
dc.publisher.placeUnited Statesen_HK
dc.identifier.scopusauthoridBai, Z=7202524223en_HK
dc.identifier.scopusauthoridYao, J=7403503451en_HK
dc.identifier.citeulike10037679-

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