Robust and efficient estimation of multivariate scatter and location

Several equivariant estimators of multivariate location and scatter are studied, which are highly robust, have a controllable finite-sample efficiency and are computationally feasible in large dimensions. The most frequently employed estimators are not quite satisfactory in this respect. The Minimum Volume Ellipsoid (MVE) and the Minimum Covariance Determinant (MCD) estimators are known to have a very low efficiency. S-estimators with a monotonic weight function like the bisquare have a low efficiency when the dimension p is small, and their efficiency tends to one with increasing p. Unfortunately, this advantage is outweighed by a serious loss in robustness for large p. Four families of estimators with controllable efficiencies whose performance for moderate to large p has not been explored to date are studied: S-estimators with a non-monotonic weight function, MM-estimators, τ-estimators, and the Stahel–Donoho estimator. Two types of starting estimators are employed: the MVE computed through subsampling, and a semi-deterministic procedure previously proposed for outlier detection, based on the projections with maximum and minimum kurtosis. A simulation study shows that an S-estimator with non-monotonic weight function can simultaneously attain high efficiency and high robustness for p≥15, while an MM-estimator with a particular weight function can be recommended for p<15. For both recommended estimators, the initial values are given by the semi-deterministic procedure mentioned above.