# Central limit theorems for eigenvalues in a spiked population model

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

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

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topBai, 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

top- [1] Z. D. Bai, B. Q. Miao and C. R. Rao. Estimation of direction of arrival of signals: Asymptotic results. Advances in Spectrum Analysis and Array Processing, S. Haykins (Ed.), vol. II, pp. 327–347. Prentice Hall’s West Nyack, New York, 1991.
- [2] Z. D. Bai. A note on limiting distribution of the eigenvalues of a class of random matrice. J. Math. Res. Exposition 5 (1985) 113–118. Zbl0591.15017MR842111
- [3] Z. D. Bai. Methodologies in spectral analysis of large dimensional random matrices, a review. Statist. Sinica 9 (1999) 611–677. Zbl0949.60077MR1711663
- [4] Z. D. Bai and J. W. Silverstein. CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32 (2004) 553–605. Zbl1063.60022MR2040792
- [5] Z. D. Bai and J. W. Silverstein. No eigenvalues outside the support of the limiting spectral distribution of large dimensional sample covariance matrices. Ann. Probab. 26 (1998) 316–345. Zbl0937.60017MR1617051
- [6] J. Baik and J. W. Silverstein. Eigenvalues of large sample covariance matrices of spiked population models. J. Multivariate Anal. 97 (2006) 1382–1408. Zbl1220.15011MR2279680
- [7] J. Baik, G. Ben Arous and S. Péché. Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33 (2005) 1643–1697. Zbl1086.15022MR2165575
- [8] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, 1985. Zbl0576.15001MR832183
- [9] I. Johnstone. On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 (2001) 295–327. Zbl1016.62078MR1863961
- [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] M. L. Mehta. Random Matrices. Academic Press, New York, 1991. Zbl0780.60014MR1083764
- [12] D. Paul. Asymptotics of the leading sample eigenvalues for a spiked covariance model. Statistica Sinica 17 (2007) 1617–1642. Zbl1134.62029MR2399865
- [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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