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Article: On sample eigenvalues in a generalized spiked population model
Title | On sample eigenvalues in a generalized spiked population model | ||||||
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Authors | |||||||
Keywords | Central limit theorems Extreme eigenvalues Largest eigenvalue Primary Sample covariance matrices Secondary Spiked population model | ||||||
Issue Date | 2012 | ||||||
Publisher | Academic 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? | ||||||
Abstract | In 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 Identifier | http://hdl.handle.net/10722/143793 | ||||||
ISSN | 2023 Impact Factor: 1.4 2023 SCImago Journal Rankings: 0.837 | ||||||
ISI Accession Number ID |
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 Field | Value | Language |
---|---|---|
dc.contributor.author | Bai, Z | en_HK |
dc.contributor.author | Yao, J | en_HK |
dc.date.accessioned | 2011-12-21T08:55:57Z | - |
dc.date.available | 2011-12-21T08:55:57Z | - |
dc.date.issued | 2012 | en_HK |
dc.identifier.citation | Journal of Multivariate Analysis, 2012, v. 106, p. 167-177 | en_HK |
dc.identifier.issn | 0047-259X | en_HK |
dc.identifier.uri | http://hdl.handle.net/10722/143793 | - |
dc.description.abstract | In 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.language | eng | en_US |
dc.publisher | Academic Press. The Journal's web site is located at http://www.elsevier.com/locate/jmva | en_HK |
dc.relation.ispartof | Journal of Multivariate Analysis | en_HK |
dc.rights | NOTICE: 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.rights | This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. | - |
dc.subject | Central limit theorems | en_HK |
dc.subject | Extreme eigenvalues | en_HK |
dc.subject | Largest eigenvalue | en_HK |
dc.subject | Primary | en_HK |
dc.subject | Sample covariance matrices | en_HK |
dc.subject | Secondary | en_HK |
dc.subject | Spiked population model | en_HK |
dc.title | On sample eigenvalues in a generalized spiked population model | en_HK |
dc.type | Article | en_HK |
dc.identifier.email | Yao, J: jeffyao@hku.hk | en_HK |
dc.identifier.authority | Yao, J=rp01473 | en_HK |
dc.description.nature | postprint | - |
dc.identifier.doi | 10.1016/j.jmva.2011.10.009 | en_HK |
dc.identifier.scopus | eid_2-s2.0-84855813409 | en_HK |
dc.identifier.hkuros | 198156 | en_US |
dc.relation.references | http://www.scopus.com/mlt/select.url?eid=2-s2.0-84855813409&selection=ref&src=s&origin=recordpage | en_HK |
dc.identifier.volume | 106 | en_HK |
dc.identifier.spage | 167 | en_HK |
dc.identifier.epage | 177 | en_HK |
dc.identifier.isi | WOS:000300913400011 | - |
dc.publisher.place | United States | en_HK |
dc.identifier.scopusauthorid | Bai, Z=7202524223 | en_HK |
dc.identifier.scopusauthorid | Yao, J=7403503451 | en_HK |
dc.identifier.citeulike | 10037679 | - |
dc.identifier.issnl | 0047-259X | - |