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Construction de bases d'ondelettes de $L^2[0,1]$ et estimation du paramètre de longue mémoire par la méthode des ondelettes.

Abstract : This study is devoted to the use of wavelets in two different fields, constructions of bases on the interval and long-memory parameter estimation. In the first part we present general constructions of orthogonal and biorthogonal multiresolution analyses on the interval. In the first one, we describe a direct method to define an orthonormal multiresolution analysis. In the second one, we use the integration and derivation method for constructing a biorthogonal multiresolution analysis. As applications, we prove that these analyses are adapted to study regular functions on the interval (H^{s}([0,1])$ et $H^{s}_{0}([0,1])$ for $s\in\mathbb{N}$). The second part is devoted to the study of adaptive wavelet-based estimators of the long-memory parameter for Gaussian then linear processes in a general semiparametric frame. We introduce and develop the choice of a data-driven optimal bandwidth. Moreover, we establish central limit theorems for the estimators of the memory parameter with the minimax rate of convergence (up to a logarithm factor). Finally adaptive goodness-of-fit tests are also built and easy to be employed: they are chi-square type tests. Simulations confirm the interesting properties of consistency and robustness of the adaptive estimators and tests.
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https://tel.archives-ouvertes.fr/tel-00666162
Contributor : Hatem Bibi Connect in order to contact the contributor
Submitted on : Friday, February 10, 2012 - 3:15:21 PM
Last modification on : Friday, May 6, 2022 - 4:50:07 PM
Long-term archiving on: : Thursday, November 22, 2012 - 11:56:44 AM

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  • HAL Id : tel-00666162, version 1

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Hatem Bibi. Construction de bases d'ondelettes de $L^2[0,1]$ et estimation du paramètre de longue mémoire par la méthode des ondelettes.. Statistiques [math.ST]. Université Panthéon-Sorbonne - Paris I, 2011. Français. ⟨tel-00666162⟩

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