Model selection for (auto-)regression with dependent data

¹ École Normale Supérieure, DMA, 45 rue d'Ulm, 75230 Paris Cedex 05, France; Yannick.Baraud@ens.fr.
² Laboratoire de Probabilités et Modèles Aléatoires, Boîte 188, Université Paris 6, 4 place Jussieu, 75252 Paris Cedex 05, France.
³ Laboratoire de Probabilités et Modèles Aléatoires, Boîte 7012, Université Paris 7, 2 place Jussieu, 75251 Paris Cedex 05, France.

Received: 15 April 1999
Revised: 20 July 1999
Revised: 14 May 2001

Abstract

In this paper, we study the problem of non parametric estimation of an unknown regression function from dependent data with sub-Gaussian errors. As a particular case, we handle the autoregressive framework. For this purpose, we consider a collection of finite dimensional linear spaces (e.g. linear spaces spanned by wavelets or piecewise polynomials on a possibly irregular grid) and we estimate the regression function by a least-squares estimator built on a data driven selected linear space among the collection. This data driven choice is performed via the minimization of a penalized criterion akin to the Mallows' C_p. We state non asymptotic risk bounds for our estimator in some -norm and we show that it is adaptive in the minimax sense over a large class of Besov balls of the form B_α,p,∞(R) with p ≥ 1.