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:**

- 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