On sample eigenvalues in a generalized spiked population model

Abstract

In the spiked population model introduced by Johnstone [10], 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 [6] 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 [5], we have provided the limiting distributions for these extreme sample eigenvalues. In this paper, we extend this theory to a 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. 1991 Mathematics Subject Classification. Primary 60F15, 60F05; secondary 15A52, 62H25.

Random matrix theory in statistics: A review

“…Benaych-Georges and Nadakuditi (2011) gave a more intuitive explanation of the phase transition phenomenon that brings out the central role played by quadratic forms involving the resolvent of a null Wishart matrix. Bai and Yao (2011) extended the scope of the results further by considering a "generalized spiked model", where the "noise" eigenvalues of the population covariance matrix decay slowly rather than remaining constant (Fig. 3).…”

Asymptotics of empirical eigenstructure for high dimensional spiked covariance

“…If the true covariance structure takes the form of a spiked matrix, Baik, Ben Arous and Péché (2005) showed that the asymptotic distribution of the top empirical eigenvalue exhibits an n 2/3 scaling when the eigenvalue lies below a threshold $1+\sqrt{\gamma }$, and an n 1/2 scaling when it is above the threshold (named BBP phase transition after the authors). The phase transition is further studied by Benaych-Georges and Nadakuditi (2011) and Bai and Yao (2012) under more general assumptions. For the case where we have the regular scaling, Paul (2007) investigated the asymptotic behavior of the corresponding empirical eigenvectors and showed that the major part of an eigenvector is normally distributed with a regular scaling n 1/2 .…”

The Isotropic Semicircle Law and Deformation of Wigner Matrices

“…See, e.g., [32] for a review of recent developments. Analogous deformations of covariance matrices, so-called spiked population models, were studied in detail in [1,2,4].…”