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

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 62F35 Robustness and adaptive procedures
- 62G05 Estimation

**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*