Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Penalized blockwise Stein's method, monotone oracles and sharp adaptive estimation

Auteur(s):

Code(s) de Classification MSC:

• 62G05 Estimation
• 62G20 Asymptotic properties

Résumé: We consider the gaussian sequence space model. Using a penalized blockwise Stein's rule with an appropriate choice of blocks and respective penalties, we construct a nonlinear estimator that enjoys simultaneously the following properties: (i) it satisfies asymptotically exact oracle inequalities within any class of linear estimates having monotone non-decreasing weights,(ii) it is sharp asymptotically minimax on any ellipsoid in $\ell_2$ with monotone non-decreasing coefficients, (iii) it is almost sharp asymptotically minimax on other bodies such as hyperrectangles, tail-classes, Besov classes with $p\ge 2$, (iv) it attains the optimal rate of convergence (up to a log-factor) on the Besov classes with $p<2$. A surprising fact is that there exists a large variety of estimators that possess these four properties simultaneously.

Mots Clés: Adaptive curve estimation ; Exact minimax constants ; Oracle inequalities ; Monotone oracle ; Penalized blockwise Stein's rule

Date: 2001-09-27

Prépublication numéro: PMA-689

Postscript file : PMA-689.ps