Issue |
ESAIM: PS
Volume 6, 2002
|
|
---|---|---|
Page(s) | 127 - 146 | |
DOI | https://doi.org/10.1051/ps:2002007 | |
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,
France; yannick.baraud@ens.fr.
Received:
28
March
2001
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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