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

### Exact adaptive pointwise estimation on Sobolev classes of densities

Auteur(s):

Code(s) de Classification MSC:

• 62G07 Curve estimation (nonparametric regression, density estimation, etc.)
• 62G20 Asymptotic properties

Résumé: The subject of this paper is to estimate adaptively the common probability density of $n$ independent, identically distributed random variables. The estimation is done at a fixed point $x_{0}\in I\!\!R$, over the density functions that belong to the Sobolev class $W_n(\beta ,L)$. We consider the adaptive problem setup, where the regularity parameter $\beta$ is unknown and varies in a given set $B_{n}$. A sharp adaptive estimator is obtained, and the explicit asymptotical constant, associated to its rate of convergence is found.

Mots Clés: Density estimation ; exact asymptotics ; pointwise risk ; sharp adaptive estimator

Date: 2000-11-16

Prépublication numéro: PMA-621