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Zbl 1059.62038
Baraud, Yannick
Model selection for regression on a random design. (English)
[J] ESAIM, Probab. Stat. 6, 127-146 (2002). ISSN 1292-8100; ISSN 1262-3318/e

Summary: We consider the problem of estimating an unknown regression function when the design is random with values in $\bbfR^k$. 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 $\cal B_{\alpha, l,\infty}(R)$ with $R>0,\ l\geq 1$ and $\alpha>\alpha_l$, where $\alpha_l$ is a positive number satisfying $1/l-1/2\leq\alpha_l<1/l$. 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 $k=1$.

MSC 2000:: *62G08 Nonparametric regression
46N30 Appl. of functional analysis in probability theory and statistics
62J02 General nonlinear regression

Keywords: least-squares estimators; penalized criteria; minimax rates; Besov spaces; model selection; adaptive estimation

Cited in: Zbl 1064.62030

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