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' Cp. 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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