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

TL;DR

This paper studies the eigenvalue behavior of high-dimensional spatial-sign covariance matrices, showing their eigenvalues converge to a generalized Marcenko-Pastur distribution and establishing a new CLT, with applications in robust high-dimensional shape analysis.

Contribution

It introduces a limit distribution and a CLT for eigenvalues of spatial-sign covariance matrices in high dimensions, with practical robust statistical applications.

Findings

Eigenvalues converge to a generalized Marcenko-Pastur distribution.

A new central limit theorem for linear spectral statistics is established.

The proposed methods improve robustness against outliers in high-dimensional data.

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.