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

In this paper, we study the problem of nonparametric estimation of the mean and variance functions b and σ2 in a model: Xi+1=b(Xi)+σ(Xi)εi+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 σ2, 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 nonasymptotic risk bounds in L2 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.