Multivariate τ‐estimators for location and scatter

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