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

### Adaptive estimation in autoregression or ß-mixing regression via model selection

Auteur(s):

Code(s) de Classification MSC:

• 62G05 Estimation
• 62J02 General nonlinear regression

Résumé: We study the problem of estimating some unknown regression function in a $\beta$-mixing dependent framework. For this end, we consider some collection of models which are finite dimensional spaces. A penalized least-squares estimator (PLSE) is built on a data driven selected model among this collection. We state non asymptotic risk bounds for this PLSE and give several examples where our procedure can be applied (autoregression, regression with arithmetically $\beta$-mixing design points, regression with mixing errors, estimation in additive frameworks, estimation of the order of the autoregression ...). In addition we show that under weak moment condition on the errors, our estimator is adaptive in the minimax sense simultaneously over some family of Besov balls.

Mots Clés: Nonparametric regression ; Least-squares estimator ; Model selection ; Adaptive estimation ; Autoregression order ; Additive framework ; Time series ; Mixing processes

Date: 2000-02-17

Prépublication numéro: PMA-566