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.
Dans un modèle de variances hétérogènes, les valeurs propres de la matrice de covariance des variables sont toutes égales à l'unité sauf un faible nombre d'entre elles. Ce modèle a été introduit par Johnstone comme une explication possible de la structure des valeurs propres de la matrice de covariance empirique constatée sur plusieurs ensembles de données réelles. Une question importante est de quantifier la perturbation causée par ces valeurs propres différentes de l'unité. Un travail récent de Baik et Silverstein établit la limite presque sûre des valeurs propres empiriques extrêmes lorsque le nombre de variables tend vers l'infini proportionnellement à la taille de l'échantillon. Ce travail établit un théorème limite central pour ces valeurs propres empiriques extrêmes. Il est basé sur un nouveau théorème limite central pour les formes sesquilinéaires aléatoires.
Keywords: sample covariance matrices, spiked population model, central limit theorems, largest eigenvalue, extreme eigenvalues, random sesquilinear forms, random quadratic forms
@article{AIHPB_2008__44_3_447_0, author = {Bai, Zhidong and Yao, Jian-Feng}, title = {Central limit theorems for eigenvalues in a spiked population model}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {447--474}, publisher = {Gauthier-Villars}, volume = {44}, number = {3}, year = {2008}, doi = {10.1214/07-AIHP118}, mrnumber = {2451053}, zbl = {1274.62129}, language = {en}, url = {http://archive.numdam.org/articles/10.1214/07-AIHP118/} }
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 SP - 447 EP - 474 VL - 44 IS - 3 PB - Gauthier-Villars UR - http://archive.numdam.org/articles/10.1214/07-AIHP118/ DO - 10.1214/07-AIHP118 LA - en ID - AIHPB_2008__44_3_447_0 ER -
%0 Journal Article %A Bai, Zhidong %A Yao, Jian-Feng %T Central limit theorems for eigenvalues in a spiked population model %J Annales de l'I.H.P. Probabilités et statistiques %D 2008 %P 447-474 %V 44 %N 3 %I Gauthier-Villars %U http://archive.numdam.org/articles/10.1214/07-AIHP118/ %R 10.1214/07-AIHP118 %G en %F AIHPB_2008__44_3_447_0
Bai, Zhidong; Yao, Jian-Feng. Central limit theorems for eigenvalues in a spiked population model. Annales de l'I.H.P. Probabilités et statistiques, Volume 44 (2008) no. 3, pp. 447-474. doi : 10.1214/07-AIHP118. http://archive.numdam.org/articles/10.1214/07-AIHP118/
[1] 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.
, and .[2] A note on limiting distribution of the eigenvalues of a class of random matrice. J. Math. Res. Exposition 5 (1985) 113-118. | MR | Zbl
.[3] Methodologies in spectral analysis of large dimensional random matrices, a review. Statist. Sinica 9 (1999) 611-677. | MR | Zbl
.[4] CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32 (2004) 553-605. | MR | Zbl
and .[5] No eigenvalues outside the support of the limiting spectral distribution of large dimensional sample covariance matrices. Ann. Probab. 26 (1998) 316-345. | MR | Zbl
and .[6] Eigenvalues of large sample covariance matrices of spiked population models. J. Multivariate Anal. 97 (2006) 1382-1408. | MR
and .[7] Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33 (2005) 1643-1697. | MR | Zbl
, and .[8] Matrix Analysis. Cambridge University Press, 1985. | MR | Zbl
and .[9] On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 (2001) 295-327. | MR | Zbl
.[10] Distribution of eigenvalues for some sets of random matrices. Math. USSR-Sb 1 (1967) 457-483. | Zbl
and .[11] Random Matrices. Academic Press, New York, 1991. | MR | Zbl
.[12] Asymptotics of the leading sample eigenvalues for a spiked covariance model. Statistica Sinica 17 (2007) 1617-1642. | MR | Zbl
.[13] A reliable data-based bandwidth selection method for kernel density estimation. J. Roy. Stat. Soc. Ser. B 53 (1991) 683-690. | MR | Zbl
and .Cited by Sources: