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On the Optimality of Multivariate S-Estimators


Christophe Croux, K. U. Leuven, Naamsestraat 69, B3000 Leuven, Belgium.


Abstract.  In this article, 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 per cent for Gaussian errors. We prove the surprising result that in dimensions larger than one, the efficiency of a maximum breakdown S-estimator of location and scatter can get arbitrarily close to 100 per cent, by an appropriate selection of the loss function.