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

### Adaptive estimation of mean and volatility functions in (auto-)regressive models

Auteur(s):

Code(s) de Classification MSC:

• 62G07 Curve estimation (nonparametric regression, density estimation, etc.)
• 62J02 General nonlinear regression

Résumé: In this paper, we study the problem of non parametric estimation of the mean and variance functions $b$ and $\sigma^2$ in a model: $X_{i+1}=b(X_i)+\sigma(X_i)\varepsilon_{i+1}$. For this purpose, we consider a collection of finite dimensional linear spaces . We estimate $b$ using a mean squares estimator built on a data driven selected linear space among the collection. Then an analogous procedure estimates $\sigma^2$, using a possibly different collection of models. Both data driven choices are performed via the minimization of penalized mean squares contrasts. The penalty functions are random in order not to depend on unknown variance-type quantities. In all cases, we state non asymptotic risk bounds in $\LL_2$ empirical norm for our estimators and we show that they are both adaptive in the minimax sense over a large class of Besov balls. Lastly, we give the results of intensive simulation experiments which show the good performances of our estimator.

Mots Clés: Nonparametric regression ; Least-squares estimator ; Adaptive estimation ; Autoregression ; Variance estimation ; Mixing processes

Date: 2001-05-03

Prépublication numéro: PMA-652

Pdf file (with figures) : PMA-652.pdf