Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

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


Code(s) de Classification MSC:

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