Central limit theorems for eigenvalues in a spiked population model

Zhidong Bai; Jian-Feng Yao

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 3, page 447-474
  • ISSN: 0246-0203

Abstract

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In a spiked population model, the population covariance matrix has all its eigenvalues equal to units except for a few fixed eigenvalues (spikes). This model is proposed by Johnstone to cope with empirical findings on various data sets. The question is to quantify the effect of the perturbation caused by the spike eigenvalues. A recent work by Baik and Silverstein establishes the almost sure limits of the extreme sample eigenvalues associated to the spike eigenvalues when the population and the sample sizes become large. This paper establishes the limiting distributions of these extreme sample eigenvalues. As another important result of the paper, we provide a central limit theorem on random sesquilinear forms.

How to cite

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Bai, Zhidong, and Yao, Jian-Feng. "Central limit theorems for eigenvalues in a spiked population model." Annales de l'I.H.P. Probabilités et statistiques 44.3 (2008): 447-474. <http://eudml.org/doc/77978>.

@article{Bai2008,
abstract = {In a spiked population model, the population covariance matrix has all its eigenvalues equal to units except for a few fixed eigenvalues (spikes). This model is proposed by Johnstone to cope with empirical findings on various data sets. The question is to quantify the effect of the perturbation caused by the spike eigenvalues. A recent work by Baik and Silverstein establishes the almost sure limits of the extreme sample eigenvalues associated to the spike eigenvalues when the population and the sample sizes become large. This paper establishes the limiting distributions of these extreme sample eigenvalues. As another important result of the paper, we provide a central limit theorem on random sesquilinear forms.},
author = {Bai, Zhidong, Yao, Jian-Feng},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {sample covariance matrices; spiked population model; central limit theorems; largest eigenvalue; extreme eigenvalues; random sesquilinear forms; random quadratic forms},
language = {eng},
number = {3},
pages = {447-474},
publisher = {Gauthier-Villars},
title = {Central limit theorems for eigenvalues in a spiked population model},
url = {http://eudml.org/doc/77978},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Bai, Zhidong
AU - Yao, Jian-Feng
TI - Central limit theorems for eigenvalues in a spiked population model
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 3
SP - 447
EP - 474
AB - In a spiked population model, the population covariance matrix has all its eigenvalues equal to units except for a few fixed eigenvalues (spikes). This model is proposed by Johnstone to cope with empirical findings on various data sets. The question is to quantify the effect of the perturbation caused by the spike eigenvalues. A recent work by Baik and Silverstein establishes the almost sure limits of the extreme sample eigenvalues associated to the spike eigenvalues when the population and the sample sizes become large. This paper establishes the limiting distributions of these extreme sample eigenvalues. As another important result of the paper, we provide a central limit theorem on random sesquilinear forms.
LA - eng
KW - sample covariance matrices; spiked population model; central limit theorems; largest eigenvalue; extreme eigenvalues; random sesquilinear forms; random quadratic forms
UR - http://eudml.org/doc/77978
ER -

References

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  8. [8] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, 1985. Zbl0576.15001MR832183
  9. [9] I. Johnstone. On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 (2001) 295–327. Zbl1016.62078MR1863961
  10. [10] V. A. Marčenko and L. A. Pastur. Distribution of eigenvalues for some sets of random matrices. Math. USSR-Sb 1 (1967) 457–483. Zbl0162.22501
  11. [11] M. L. Mehta. Random Matrices. Academic Press, New York, 1991. Zbl0780.60014MR1083764
  12. [12] D. Paul. Asymptotics of the leading sample eigenvalues for a spiked covariance model. Statistica Sinica 17 (2007) 1617–1642. Zbl1134.62029MR2399865
  13. [13] S. J. Sheather and M. C. Jones. A reliable data-based bandwidth selection method for kernel density estimation. J. Roy. Stat. Soc. Ser. B 53 (1991) 683–690. Zbl0800.62219MR1125725

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