Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Model selection via testing: an alternative to (penalized) maximum likelihood estimators


Code(s) de Classification MSC:

Résumé: This paper is devoted to the description and study of a family of estimators, that we shall call $T$-estimators ($T$ for tests), for minimax estimation and model selection. Their construction is based on former ideas about deriving estimators from some families of tests due to Le Cam (1973 and 1975) and Birg\'e (1983, 1984a and b) and about complexity based model selection from Barron and Cover (1991). It is well-known that maximum likelihood estimators or, more generally, minimum contrast estimators do suffer from various weaknesses, and their penalized versions as well. In particular they are not robust and they require restrictive assumptions on both the models and the underlying parameter for the estimators to work correctly. Our method, which derives an estimator from many simultaneous tests between some probability balls in a suitable metric space, tends to solve many of these difficulties. Its robustness properties allow to deal with minimax estimation and model selection in a unified way, since bounding the minimax risk amounts to perform the method with a single, well-chosen, model. This results in simple bounds for the minimax risk solely based on some metric properties of the parameter space. Moreover the method applies to various statistical frameworks (we shall concentrate here on the i.i.d.\ and the Gaussian sequence settings only) and can handle essentially all types of models, linear or not, parametric and non-parametric, simultaneously. From these viewpoints, it is much more flexible than traditional methods. In particular, we shall be able to derive some adaptation results for density estimation over Besov balls that do not seem to be accessible to classical methods. The counterpart for these nice properties is the very high computational complexity of our construction which makes our estimators look more like theoretical than practical tools.

Mots Clés: Maximum likelihood ; robustness ; robust tests ; metric dimension ;l minimax risk ; model selection ; aggregation of estimators

Date: 2003-11-05

Prépublication numéro: PMA-862