Multivariate TAU-estimators for location and scatter

Abstract

We discuss the robustness and asymptotic behaviour of tau-estimators for multivariate location and scatter. We show that tau-estimators correspond to multivariate M-estimators defined by a weighted average of redescending psi-functions, where the weights are adaptive. We prove consistency and asymptotic normality under weak assumptions on the underlying distribution, show that tau-estimators have a high breakdown point, and obtain the influence function at general distributions. In the special case of a location-scatter family, tau-estimators are asymptotically equivalent to multivariate S-estimators defined by means of a weighted rho-function. This enables us to combine a high breakdown point and bounded influence with good asymptotic efficiency for the location and covariance estimator.