Estimators Asymptotically Minimax in Wide Sense

Abstract Estimators of location are considered. Huber (1964) introduced estimators asymptotically minimax on the set 𝔖 of all regular M‐estimators, for a given contamination ε and for the set Q of all regular symmetric alternative data sources. We extend his concept by admitting arbitrary sets 𝔖 of regular M‐estimators and arbitrary sets Q or regular symmetric alternative sources, and also by replacing the singletons [ε] ⊂ (0, 1) by arbitrary subsets 𝔖 ⊂ (0, 1). The resulting estimator cannot in general be evaluated explicitly. But for finite T it exists and, if 𝔖 and Q are finite too, it may be chosen by a computer. This extra burden is justified in some cases since more than 100% relative efficiency gain against all Huber's Hk is achievable in this manner. Such gains are achieved for a nontrivial family Q by the estimator proposed in Vajda (1984), with redescending influence curve, which is shown to be asymptotically minimax in wide sense.