Abstract.
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.
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Received: 2 July 1997 / Revised version: 20 September 1999 / Published online: 6 July 2000
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Baraud, Y. Model selection for regression on a fixed design. Probab Theory Relat Fields 117, 467–493 (2000). https://doi.org/10.1007/PL00008731
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DOI: https://doi.org/10.1007/PL00008731
- Key words and phrases: Nonparametric regression – Least-squares estimator – Model selection – Adaptive estimation – Moment inequality – Concentration of measure – Empirical processes
- Mathematics Subject Classification (1991): Primary 62G07; Secondary 62J02, 60E15