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:**

- 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*