TY - JOUR

T1 - Limiting laws for extreme eigenvalues of large-dimensional spiked Fisher matrices with a divergent number of spikes

AU - Xie, Junshan

AU - Zeng, Yicheng

AU - ZHU, Lixing

N1 - Funding Information:
The authors gratefully acknowledge a grant from the University Grants Council of Hong Kong and a NSFC, China grant (NSFC11671042). Drs Xie and Zeng are co-first authors. Zeng and Zhu are in charge of all revisions. The authors thank Editor, Associate editor and two referees for their constructive comments and suggestions that led to an improvement of the early manuscript.

PY - 2021/7

Y1 - 2021/7

N2 - Consider the p×p matrix that is the product of a population covariance matrix and the inverse of another population covariance matrix. Suppose that their difference has a divergent rank with respect to p, when two samples of sizes n and T from the two populations are available, we construct its corresponding sample version. In the high-dimensional regime where both n and T are proportional to p, we investigate the limiting laws for extreme (spiked) eigenvalues of the sample (spiked) Fisher matrix when the number of spikes is divergent and these spikes are unbounded. We derive the convergence in probability of these spiked eigenvalues after scaling, and the central limit theorem for normalized spiked eigenvalues.

AB - Consider the p×p matrix that is the product of a population covariance matrix and the inverse of another population covariance matrix. Suppose that their difference has a divergent rank with respect to p, when two samples of sizes n and T from the two populations are available, we construct its corresponding sample version. In the high-dimensional regime where both n and T are proportional to p, we investigate the limiting laws for extreme (spiked) eigenvalues of the sample (spiked) Fisher matrix when the number of spikes is divergent and these spikes are unbounded. We derive the convergence in probability of these spiked eigenvalues after scaling, and the central limit theorem for normalized spiked eigenvalues.

KW - Extreme eigenvalue

KW - Fisher matrix

KW - Phase transition phenomenon

KW - Random matrix theory

KW - Spiked population model

UR - http://www.scopus.com/inward/record.url?scp=85103011196&partnerID=8YFLogxK

U2 - 10.1016/j.jmva.2021.104742

DO - 10.1016/j.jmva.2021.104742

M3 - Article

AN - SCOPUS:85103011196

SN - 0047-259X

VL - 184

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

M1 - 104742

ER -