On the Optimality of Multivariate S-Estimators

In this paper we maximize the efficiency of a multivariate S-estimator under a constraint on the breakdown point. In the linear regression model, it is known that the highest possible efficiency of a maximum breakdown S-estimator is bounded above by 33% for Gaussian errors. We prove the surprising result that in dimensions larger than one, the efficiency of a maxi- mum breakdown S-estimator of location and scatter can get arbitrarily close to 100%, by an appropriate selection of the loss function.