The consistency and asymptotic normality of the spatial sign covariance matrix with unknown location are shown. Simulations illustrate the different asymptotic behavior when using the mean and the spatial median as location estimator.2010 MSC: 62H12, 62G20, 62H11

A new robust correlation estimator based on the spatial sign covariance matrix (SSCM) is proposed. We derive its asymptotic distribution and influence function at elliptical distributions. Finite sample and robustness properties are studied and compared to other robust correlation estimators by means of numerical simulations.

We gather several results on the eigenvalues of the spatial sign covariance matrix of an elliptical distribution. It is shown that the eigenvalues are a one-to-one function of the eigenvalues of the shape matrix and that they are closer together than the latter. We further provide a one-dimensional integral representation of the eigenvalues, which facilitates their numerical computation.

The autocorrelation function (acf) and the partial autocorrelation function (pacf) are elementary tools of linear time series analysis. The sensitivity of the conventional sample acf and pacf to outliers is well known. We review robust estimators and evaluate their performances in different data situations considering Gaussian scenarios with and without outliers in a simulation study.

We summarize properties of the spatial sign covariance matrix and especially consider the relationship between its eigenvalues and those of the shape matrix of an elliptical distribution. The explicit relationship known in the bivariate case was used to construct the spatial sign correlation coefficient, which is a non-parametric and robust estimator for the correlation coefficient within the elliptical model. We consider a multivariate generalization, which we call the multivariate spatial sign correlation matrix. A small simulation study indicates that the new estimator is very efficient under various elliptical distributions if the dimension is large. We furthermore derive its influence function under certain conditions which indicates that the multivariate spatial sign correlation becomes more sensitive to outliers as the dimension increases.