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Model selection for (auto-)regression with dependent data

Published online by Cambridge University Press:  15 August 2002

Yannick Baraud
École Normale Supérieure, DMA, 45 rue d'Ulm, 75230 Paris Cedex 05, France;
F. Comte
Laboratoire de Probabilités et Modèles Aléatoires, Boîte 188, Université Paris 6, 4 place Jussieu, 75252 Paris Cedex 05, France.
G. Viennet
Laboratoire de Probabilités et Modèles Aléatoires, Boîte 7012, Université Paris 7, 2 place Jussieu, 75251 Paris Cedex 05, France.
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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' Cp. We state non asymptotic risk bounds for our estimator in some ${\mathbb{L}}_2$-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.

Research Article
© EDP Sciences, SMAI, 2001

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