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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Abstract. The aim of this paper is to present a new estimation procedure that can be applied in various statistical frameworks including density and regression and which leads to both robust and optimal (or nearly optimal) estimators. In density estimation, they asymptotically coincide with the celebrated maximum likelihood estimators at least when the statistical model is regular enough and contains the true density to estimate. For very general models of densities, including non-compact ones, these estimators are robust with respect to the Hellinger distance and converge at optimal rate (up to a possible logarithmic factor) in all cases we know. In the regression setting, our approach improves upon the classical least squares in many respects. In simple linear regression for example, it provides an estimation of the coefficients that are both robust to outliers and simultaneously rateoptimal (or nearly rate-optimal) for a large class of error distributions including Gaussian, Laplace, Cauchy and uniform among others.

Let $Y$ be a Gaussian vector whose components are independent with a common unknown variance. We consider the problem of estimating the mean $\mu$ of $Y$ by model selection. More precisely, we start with a collection $\mathcal{S}=\{S_m,m\in\mathcal{M}\}$ of linear subspaces of $\mathbb{R}^n$ and associate to each of these the least-squares estimator of $\mu$ on $S_m$. Then, we use a data driven penalized criterion in order to select one estimator among these. Our first objective is to analyze the performance of estimators associated to classical criteria such as FPE, AIC, BIC and AMDL. Our second objective is to propose better penalties that are versatile enough to take into account both the complexity of the collection $\mathcal{S}$ and the sample size. Then we apply those to solve various statistical problems such as variable selection, change point detections and signal estimation among others. Our results are based on a nonasymptotic risk bound with respect to the Euclidean loss for the selected estimator. Some analogous results are also established for the Kullback loss.Comment: Published in at http://dx.doi.org/10.1214/07-AOS573 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org