On eigenvalues of a high-dimensional spatial-sign covariance matrix

Weiming Li; Qinwen Wang; Jianfeng Yao; Wang Zhou

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arXiv · 1901.10549

On eigenvalues of a high-dimensional spatial-sign covariance matrix

Weiming Li·Qinwen Wang·Jianfeng Yao·Wang Zhou

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Abstract

This paper investigates limiting properties of eigenvalues of multivariate sample spatial-sign covariance matrices when both the number of variables and the sample size grow to infinity. The underlying p-variate populations are general enough to include the popular independent components model and the family of elliptical distributions. A first result of the paper establishes that the distribution of the eigenvalues converges to a deterministic limit that belongs to the family of generalized Marcenko-Pastur distributions. Furthermore, a new central limit theorem is established for a class of linear spectral statistics. We develop two applications of these results to robust statistics for a high-dimensional shape matrix. First, two statistics are proposed for testing the sphericity. Next, a spectrum-corrected estimator using the sample spatial-sign covariance matrix is proposed. Simulation experiments show that in high dimension, the sample spatial-sign covariance matrix provides a valid and robust tool for mitigating influence of outliers.

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BibTeX RIS

Weiming Li, Qinwen Wang, Jianfeng Yao, Wang Zhou. 2021-01-22. On eigenvalues of a high-dimensional spatial-sign covariance matrix. https://arxiv.org/abs/1901.10549

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