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Probability Theory and Related Fields

, Volume 117, Issue 4, pp 467–493 | Cite as

Model selection for regression on a fixed design

  • Yannick Baraud

Abstract.

We deal with the problem of estimating some unknown regression function involved in a regression framework with deterministic design points. For this end, we consider some collection of finite dimensional linear spaces (models) and the least-squares estimator built on a data driven selected model among this collection. This data driven choice is performed via the minimization of some penalized model selection criterion that generalizes on Mallows' C p . We provide non asymptotic risk bounds for the so-defined estimator from which we deduce adaptivity properties. Our results hold under mild moment conditions on the errors. The statement and the use of a new moment inequality for empirical processes is at the heart of the techniques involved in our approach.

Key words and phrases: Nonparametric regression – Least-squares estimator – Model selection – Adaptive estimation – Moment inequality – Concentration of measure – Empirical processes 
Mathematics Subject Classification (1991): Primary 62G07; Secondary 62J02, 60E15 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Yannick Baraud
    • 1
  1. 1.DMA, Ecole Normale Supérieure, 45 rue d'Ulm, 75230 Paris Cedex 05, France. e-mail: yannick.baraud@ens.frFR

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