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Estimation récursive de fonctionnelles

Abstract : The aim of this thesis is the study of the asymptotic behaviour of the kernel estimator of a probability density function and its derivatives, of a regression function, as well as of the location and of the size of the mode of a probability density. The goal is to establish several properties of the recursive or semi-recursive kernel estimators in order to compare their asymptotic behaviour with that of the classical estimators. In the first chapter, we establish a large deviations principle (LDP) and a moderate deviations principle (MDP) for the recursive estimator of a probability density and for its derivatives. It turns out that, in the deviations principles for the derivatives estimators, the rate function is always quadratic, the deviations being either large or moderate. On the other hand, for the density estimator, the rate function which appears is of different nature according to whether the deviations are large or moderate. The rate functions which appear in the LDP for the derivatives and in the MDP for the density and its derivatives are larger in the case the recursive estimator is used. In the second chapter, we establish LDP and MDP for kernel estimators of the regression. We generalize the results already obtained in the unidimensional case for the Nadaraya-Watson estimator. We then study the behaviour in deviations of the semi-recursive version of this estimator by establishing a LDP and MDP. The rate function which appears in the MDP are larger for the semi-recursive estimator than for the classical estimator. In the third chapter, we are interested in the joint estimation of the location and of the size of the mode of a probability density based on the recursive kernel density estimator. We study the weak and almost sure convergence rates of the couple formed by these two estimators. To estimate the two parameters simultaneously in an optimal way, it is necessary to use different bandwidths to define each of the two estimators. The semi-recursive estimators lead to asymptotic variances smaller than the classical estimators.
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Contributor : Baba Thiam <>
Submitted on : Thursday, February 15, 2007 - 2:51:35 PM
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  • HAL Id : tel-00131199, version 1



Baba Thiam. Estimation récursive de fonctionnelles. Mathématiques [math]. Université de Versailles-Saint Quentin en Yvelines, 2006. Français. ⟨tel-00131199⟩



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