Central limit theorems for eigenvalues in a spiked population model

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.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

The Isotropic Semicircle Law and Deformation of Wigner Matrices

“…For an N N matrix A we denote by 1 .A/ 6 6 N .A/ the nondecreasing sequence of eigenvalues of A. Moreover, we denote by .A/ the spectrum of A.…”

“…Next, we identify the asymptotic laws of 1 ; : : : ; 6 . There is nothing to be done with 1 . By definition,…”

On Determining the Number of Spikes in a High-Dimensional Spiked Population Model

“…Nevertheless, our method still works: we can explain it by the fact that the convergence of the λ n,i , for i > q 0 (non-spikes) is in O P 1 n 2/3 , whereas that of the difference corresponding of two identical spikes is in O P 1 √ n (Consequence of theorem 3.1 of Bai & Yao [7]). Furthermore, the variance in the convergence of this difference is 2α…”

“…For the proof, we need two theorems. The first, Proposition 3.1, shows that the limiting law of λ n,j − φ(α j ) is Gaussian (Bai and Yao [7] and Paul [8]):…”

“…• Model 4: q 0 = 6, (α 1 , α 2 , α 3 , α 4 , α 5 , α 6 ) = (7,6,6,6,5,4). For each model, two different values of c, 0.3 and 0.6, are considered, and we give in Figure 5 the frequency ofq n = q 0 and in Table 12 the mean and the mean squared error of our estimator over 1000 independent replications.…”