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Estimators Asymptotically Minimax in Wide Sense
Author(s) -
Vajda Igor
Publication year - 1989
Publication title -
biometrical journal
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.108
H-Index - 63
eISSN - 1521-4036
pISSN - 0323-3847
DOI - 10.1002/bimj.4710310706
Subject(s) - minimax , estimator , mathematics , set (abstract data type) , minimax estimator , asymptotically optimal algorithm , finite set , mathematical optimization , combinatorics , statistics , computer science , mathematical analysis , minimum variance unbiased estimator , programming language
Estimators of location are considered. Huber (1964) introduced estimators asymptotically minimax on the set 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 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 ⊂ (0, 1). The resulting estimator cannot in general be evaluated explicitly. But for finite T it exists and, if 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.