On eigenvalues of a high-dimensional spatial-sign covariance matrix
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Sample spatial-sign covariance matrix is a much-valued alternative to sample covariance matrix in robust statistics to mitigate influence of outliers. Although this matrix is widely studied in the literature, almost nothing is known on its properties when the number of variables becomes large as compared to the sample size. This paper for the first time investigates the large-dimensional limits of the eigenvalues of a sample spatial sign matrix when both the dimension and the sample size tend to infinity. A first result of the paper establishes that the distribution of the eigenvalues converges to a deterministic limit that belongs to the family of celebrated generalized Marčenko-Pastur distributions. Using tools from random matrix theory, we further establish a new central limit theorem for a general class of linear statistics of these sample eigenvalues. In particular, asymptotic normality is established for sample spectral moments under mild conditions. This theory is established when the population is elliptically distributed. As applications, we first develop two new tools for estimating the population eigenvalue distribution of a large spatial sign covariance matrix, and then for testing the order of such population eigenvalue distribution when these distributions are finite mixtures. Using these inference tools and considering the problem of blind source separation, we are able to show by simulation experiments that in high-dimensional situations, the sample spatial-sign covariance matrix is still a valid and much better alternative to sample covariance matrix when samples contain outliers.
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