Weighted Statistic in Detecting Faint and Sparse Alternatives for High-Dimensional Covariance Matrices

ABSTRACT

This article considers testing equality of two population covariance matrices when the data dimension p diverges with the sample size n (p/n → c > 0). We propose a weighted test statistic that is data-driven and powerful in both faint alternatives (many small disturbances) and sparse alternatives (several large disturbances). Its asymptotic null distribution is derived by large random matrix theory without assuming the existence of a limiting cumulative distribution function of the population covariance matrix. The simulation results confirm that our statistic is powerful against all alternatives, while other tests given in the literature fail in at least one situation. Supplementary materials for this article are available online.

KEYWORDS:

• Empirical spectral distribution
• Faint alternative
• Large random matrix theory
• Linear spectral statistic
• Sparse alternative
• Stieltjes transform

Supplementary Materials

The online supplementary files contain the appendices for the article.

Acknowledgments

The authors thank the editor, the associate editor, and the referees for their constructive comments that substantially improved the article.

Funding

G. M. Pan (corresponding author) was partially supported by a MOE Tier 2 grant 2014-T2-2-060 and by a MOE Tier 1 Grant RG25/14 at the Nanyang Technological University, Singapore.