Perforamances statistiques d'estimateurs non-linéaires

Abstract : We work in the context of nonparametric estimation in the regression model. Firstly, we consider observations Y where the density $g$ is known and depends on a regression function $f (X)$ unknown. In this thesis, this function is assumed regular, i.e. belonging to a Hölder ball. The goal is to estimate the function $f$ to a point $y$ (pointwise estimation). For it, we develop a {\it local bayesian estimator}, constructed from the density $g$ of the observations. We propose an adaptive procedure based on the Lepski's method, which allows to construct an adaptive estimator chosen from the family of {\it local bayesian estimators} indexed by the bandwidth. Under some sufficient assumptions on the density $g$, our estimator achieves the adaptive optimal rate (in a particular sense). In addition, we remark that, in some models, the bayesian estimator is more efficient than linear estimators. Secondly, another approach is considered. We consider the additive regression model, where the density of the noise is unknown and assumed to be symmetric. In this framework, we develop the so-called {\it Huber estimator} based on the idea of the median ($\ell_1$ criterion). This estimator allows to estimate the regression function, for any density of the additive noise (for example : Gaussian density or Cauchy density). With the Lepski's method, we select an estimator that achieves the rate of conventional adaptive linear estimators on Hölder spaces without depends on the symmetric density of the noise.
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https://tel.archives-ouvertes.fr/tel-00540963
Contributor : Michael Chichignoud <>
Submitted on : Monday, November 29, 2010 - 3:26:10 PM
Last modification on : Wednesday, October 10, 2018 - 1:26:43 AM
Long-term archiving on : Friday, December 2, 2016 - 8:09:50 PM

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• HAL Id : tel-00540963, version 1

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Michael Chichignoud. Perforamances statistiques d'estimateurs non-linéaires. Mathématiques [math]. Université de Provence - Aix-Marseille I, 2010. Français. ⟨tel-00540963⟩

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