In the spiked population model introduced by Johnstone (2001),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) 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), we have provided the limiting distributions for these extreme sample eigenvalues. In this paper, we extend this theory to a {\em generalized} spiked population model where the base population covariance matrix is arbitrary, instead of the identity matrix as in Johnstone's case. New mathematical tools are introduced for establishing the almost sure convergence of the sample eigenvalues generated by the spikes.
Publié le : 2008-06-06
Classification:
Extreme eigenvalues,
Sample covariance matrices,
Spiked population model,
Central limit theorems,
Largest eigenvalue,
Extreme eigenvalues.,
MSC Primary 60F15, 60F05; secondary 15A52, 62H25 62H25,
[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST],
[STAT.TH]Statistics [stat]/Statistics Theory [stat.TH],
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
@article{hal-00284468,
author = {Bai, Zhidong and Yao, Jian-Feng},
title = {Limit theorems for sample eigenvalues in a generalized spiked population model},
journal = {HAL},
volume = {2008},
number = {0},
year = {2008},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00284468}
}
Bai, Zhidong; Yao, Jian-Feng. Limit theorems for sample eigenvalues in a generalized spiked population model. HAL, Tome 2008 (2008) no. 0, . http://gdmltest.u-ga.fr/item/hal-00284468/