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Model selection for regression on a fixed design. (English) Zbl 0997.62027
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.

62G08Nonparametric regression
62G20Nonparametric asymptotic efficiency
60E15Inequalities in probability theory; stochastic orderings
62J02General nonlinear regression
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