Redescending M-estimators

In finite sample studies redescending M-estimators outperform bounded M-estimators (see for example, Andrews et al. [1972. Robust Estimates of Location. Princeton University Press, Princeton]). 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 study 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 over a class of densities the minimum variance sensitivity over a class of estimators. As a particular result, we prove that Smith's estimator, which is a compromise between Huber's skipped mean and Tukey's biweight, provides a guaranteed level of an estimator's variance sensitivity over the class of densities with a bounded variance.