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

### Adaptive estimation of the stationary density of discrete and continuous time mixing processes

Auteur(s):

Code(s) de Classification MSC:

• 62G07 Curve estimation (nonparametric regression, density estimation, etc.)
• 62M99 None of the above but in this section

Résumé: In this paper, we study the problem of non parametric estimation of the stationary density $f$ of an $\alpha$ or a $\beta$-mixing process, observed either in continuous time or in discrete time. We present an unified framework allowing to deal with many different cases. We consider a collection of finite dimensional linear regular spaces. We estimate $f$ using a projection estimator built on a data driven selected linear space among the collection. This data driven choice is performed via the minimization of a penalized contrast. We state non asymptotic risk bounds in $\mathbb{L}_2$ norm for all our estimators and in both cases of mixing. We show that they are adaptive in the minimax sense over a large class of Besov balls. In discrete time, we provide a result for model selection among an exponentially large collection of models (non regular case).

Mots Clés: Nonparametric estimation ; Projection estimator ; Adaptive estimation ; Model selection ; Mixing processes ; Continuous time ; Discrete time

Date: 2001-07-04

Prépublication numéro: PMA-678