Model selection for regression on a random design
ESAIM: Probability and Statistics, Tome 6 (2002), pp. 127-146.

We consider the problem of estimating an unknown regression function when the design is random with values in ℝ k . 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 ℬ α,l,∞ (R) with R>0, l≥1 and α>α l where α l is a positive number satisfying 1/l-1/2≤α l <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.

DOI : 10.1051/ps:2002007
Classification : 62G07, 62J02
Mots-clés : nonparametric regression, least-squares estimators, penalized criteria, minimax rates, Besov spaces, model selection, adaptive estimation 
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     pages = {127--146},
     publisher = {EDP-Sciences},
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     doi = {10.1051/ps:2002007},
     mrnumber = {1918295},
     zbl = {1059.62038},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps:2002007/}
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Baraud, Yannick. Model selection for regression on a random design. ESAIM: Probability and Statistics, Tome 6 (2002), pp. 127-146. doi : 10.1051/ps:2002007. http://www.numdam.org/articles/10.1051/ps:2002007/

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