Baraud, Yannick Model selection for regression on a fixed design. (English) Zbl 0997.62027 Probab. Theory Relat. Fields 117, No. 4, 467-493 (2000). Summary: 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’ \(C_p\). 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. Cited in 1 ReviewCited in 46 Documents MSC: 62G08 Nonparametric regression and quantile regression 62G20 Asymptotic properties of nonparametric inference 60E15 Inequalities; stochastic orderings 62J02 General nonlinear regression Keywords:adaptive estimation; moment inequality; concentration of measure; empirical processes; least-squares estimator PDFBibTeX XMLCite \textit{Y. Baraud}, Probab. Theory Relat. Fields 117, No. 4, 467--493 (2000; Zbl 0997.62027) Full Text: DOI