166 research outputs found
Limit theorems for sample eigenvalues in a generalized spiked population model
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.Comment: 24 pages; 4 figure
Central limit theorems for eigenvalues in a spiked population model
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.Comment: Published in at http://dx.doi.org/10.1214/07-AIHP118 the Annales de
l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques
(http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics
(http://www.imstat.org
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