The spatial sign covariance matrix and its application for robust correlation estimation

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.