Application des méthodes d'approximations stochastiques à l'estimation de la densité et de la régression

Abstract : The objective of this thesis is to apply the stochastic approximations methods to the estimation of a density and of a regression function. In the first chapter, we build up a stochastic algorithm with single stepsize, which defines a whole family of recursive kernel estimators of a probability density. We study the properties of this algorithm. In particular, we identify two classes of estimators; the first one corresponds to a choice of stepsize which allows to get a minimum mean squared error, the second one a minimum variance. In the second chapter, we consider the estimator proposed by Révész (1973, 1977) to estimate a regression function r:x->E[Y|X=x\]. His estimator r_n, built up by using a single-time-scale stochastic algorithm, has a big disadvantage: the assumptions on the marginal density of X necessary to establish the convergence rate of r_n are much stronger than those usually required to study the asymptotic behavior of an estimator of a regression function. We show how the application of the averaging principle of stochastic algorithms allows, by first generalizing the definition of the estimator of Révész and then by averaging this generalized estimator, to build up a recursive estimator br_n which has good asymptotic properties. In the third chapter, we still apply stochastic approximation methods to estimate a regression function. But this time, rather than to use single-time-scale stochastic algorithm, we show how the two-time-scale stochastic algorithms allow to build up a whole class of recursive estimators of a regression function, and we study the asymptotic properties of these estimators. This approach is much easier than the one of the second chapter: the estimators built up using the two-time-scale algorithms do not need to be averaged to have good asymptotic properties.
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https://tel.archives-ouvertes.fr/tel-00131964
Contributor : Yousri Slaoui <>
Submitted on : Monday, February 19, 2007 - 5:29:58 PM
Last modification on : Wednesday, January 23, 2019 - 2:39:26 PM
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Yousri Slaoui. Application des méthodes d'approximations stochastiques à l'estimation de la densité et de la régression. Mathématiques [math]. Université de Versailles-Saint Quentin en Yvelines, 2006. Français. ⟨tel-00131964⟩

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