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

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