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Model selection for (auto-)regression with dependent data

Published online by Cambridge University Press:  15 August 2002

Yannick Baraud ,
F. Comte  and
G. Viennet
Yannick Baraud
Affiliation:
École Normale Supérieure, DMA, 45 rue d'Ulm, 75230 Paris Cedex 05, France; [email protected].
F. Comte
Affiliation:
Laboratoire de Probabilités et Modèles Aléatoires, Boîte 188, Université Paris 6, 4 place Jussieu, 75252 Paris Cedex 05, France.
G. Viennet
Affiliation:
Laboratoire de Probabilités et Modèles Aléatoires, Boîte 7012, Université Paris 7, 2 place Jussieu, 75251 Paris Cedex 05, France.
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Abstract

In this paper, we study the problem of non parametric estimationof an unknown regression function from dependent data withsub-Gaussian errors. As a particular case, we handle theautoregressive framework. For this purpose, we consider acollection of finite dimensional linear spaces (e.g. linear spacesspanned by wavelets or piecewise polynomials on a possiblyirregular grid) and we estimate the regression function by aleast-squares estimator built on a data driven selected linearspace among the collection. This data driven choice is performedvia the minimization of a penalized criterion akin to the Mallows'C p . We state non asymptotic risk bounds for our estimator insome ${\mathbb{L}}_2$ -norm and we show that it is adaptive in the minimaxsense over a large class of Besov balls of the form Bα,p,∞(R) with p ≥ 1.

Keywords

Nonparametric regressionleast-squares estimatoradaptive estimationautoregressionmixingprocesses.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 5 , 2001 , pp. 33 - 49
Copyright
© EDP Sciences, SMAI, 2001

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