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Estimation adaptative par sélection de partitions en rectangles dyadiques

Abstract : In this thesis, we study several estimation problems by selection of a best piecewise constant or piecewise polynomial estimator built on a partition into dyadic intervals or rectangles, using an adequate least-squares type criterion. Our works are devoted to three topics. First, we are concerned with discrete distribution estimation and provide an application to multiple change-point detection. Then, we propose a unified approach to functional estimation problems based on possibly censored data. Last, we lead a simultaneous study of multivariate density and conditional density estimation based on dependent data. The choice of the collection of partitions into dyadic intervals or rectangles reveals highly interesting in theory and in practice. As a matter of fact, our penalized estimator satisfies in each framework a nonasymptotic oracle-type inequality for a well-chosen penalty. It also reaches the minimax rate, up to a constant, over a wide range of classes of functions that may have inhomogeneous and anisotropic smoothness. Such a property, that, up to our knowledge, has never been proved for any other estimator, follows from approximation results whose proofs are inspired from a paper by DeVore and Yu. Besides, in a univariate framework, our estimator can be determined via a shortest-path algorithm whose computational complexity is only linear in the sample size.
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Contributor : Nathalie Akakpo Connect in order to contact the contributor
Submitted on : Wednesday, January 20, 2010 - 4:26:27 AM
Last modification on : Wednesday, October 14, 2020 - 4:00:15 AM
Long-term archiving on: : Friday, June 18, 2010 - 1:07:59 AM


  • HAL Id : tel-00448753, version 1



Nathalie Akakpo. Estimation adaptative par sélection de partitions en rectangles dyadiques. Mathématiques [math]. Université Paris Sud - Paris XI, 2009. Français. ⟨tel-00448753⟩



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