Volume 6, 2002
|Page(s)||127 - 146|
|Published online||15 November 2002|
Model selection for regression on a random design
École Normale Supérieure, DMA, 45 rue d'Ulm, 75230 Paris Cedex 05,
Revised: 5 May 2002
We consider the problem of estimating an unknown regression function when the design is random with values in . Our estimation procedure is based on model selection and does not rely on any prior information on the target function. We start with a collection of linear functional spaces and build, on a data selected space among this collection, the least-squares estimator. We study the performance of an estimator which is obtained by modifying this least-squares estimator on a set of small probability. For the so-defined estimator, we establish nonasymptotic risk bounds that can be related to oracle inequalities. As a consequence of these, we show that our estimator possesses adaptive properties in the minimax sense over large families of Besov balls Bα,l,∞(R) with R>0, l ≥ 1 and α > α1 where α1 is a positive number satisfying 1/l - 1/2 ≤ α1 < 1/l. We also study the particular case where the regression function is additive and then obtain an additive estimator which converges at the same rate as it does when k=1.
Mathematics Subject Classification: 62G07 / 62J02
Key words: Nonparametric regression / least-squares estimators / penalized criteria / minimax rates / Besov spaces / model selection / adaptive estimation.
© EDP Sciences, SMAI, 2002
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