Model selection for (auto-)regression with dependent data

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 Bα,p,∞ (R) with p ≥ 1.