Redescending M -estimators

In finite sample studies redescending M -estimators outperform bounded M -estimators (see for example, Andrews et al., 1972). Even though redescenders arise naturally out of the maximum likelihood approach if one uses very heavy-tailed models, the commonly used redescenders have been derived from purely heuristic considerations. Using a recent approach proposed by Shurygin, we studied the optimality of redescending M -estimators. We show that redescending M -estimator can be designed by applying a global minimax criterion to locally robust estimators, namely maximizing the minimum variance sensitivity of an estimator over a given class of densities. As a particular result, we proved that Smith’s estimator, which is a compromise between Huber’s skipped mean and Tukey’s biweight, provides the guaranteed level of an estimator’s variance sensitivity over the class of distribution densities with a bounded variance. AMS Subject Classification: 62G35