Bootstrap adaptive estimation: The trimmed-mean example

We consider the problem of choosing among a class of possible estimators by selecting the estimator with the smallest bootstrap estimate of finite sample variance. This is an alternative to using cross-validation to choose an estimator adaptively. The problem of a confidence interval based on such an adaptive estimator is considered. We illustrate the ideas by applying the method to the problem of choosing the trimming proportion of an adaptive trimmed mean. It is shown that a bootstrap adaptive trimmed mean is asymptotically normal with an asymptotic variance equal to the smallest among trimmed means. The asymptotic coverage probability of a bootstrap confidence interval based on such adaptive estimators is shown to have the nominal level. The intervals based on the asymptotic normality of the estimator share the same asymptotic result, but have poor small-sample properties compared to the bootstrap intervals. A small-sample simulation demonstrates that bootstrap adaptive trimmed means adapt themselves rather well even for samples of size 10. Nous etudions le probleme du choix d'un estimateur parmi une classe en selectionnant celui qui a la plus petite estimation bootstrap de la variance d'echantillon fini. Ceci est une alternative a l'utilisation de la validation croisee pour un choix adaptatif d'estimateur. On considere le probleme de la construction d'intervalles de confiance pour de tels estimateurs adaptables. Les idees sont illustrees en appliquant la methode au probleme du choix de la proportion de troncature d'une moyenne tronquee adaptative. Il est demontre qu'une moyenne tronquee adaptative bootstrap est asymptotiquement normale avec une variance asymptotique egale a la plus petite parmi celles des moyennes tronquees. La probabilite de couverture asymptotique d'un intervalle de confiance bootstrap base sur de tels estimateurs adaptables est egale a la probabilite prescrite. Les intervalles bases sur la normalite asymptotique ont la meme propriete, mais ils ont une bien plus faible performance que les intervalles bootstrap dans le cas de petits echantillons. Les resultats d'une simulation demontrent que les moyennes tronquees adaptatives bootstrap s'adaptent tres bien, meme pour des echantillons de taille 10.

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