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Theses

Estimation bayésienne non paramétrique

Abstract : In the framework of a wavelet analysis, we study the statistical meaning of a special class of Lorentz spaces: the weak Besov spaces are naturally appearing in the maxiset theory. With 'Gaussian white noise' type assumptions, we show, using Bayesian tools, that the minimax rates associated with strong or weak Besov spaces are the same. We exhibit the least favorable priors for each weak Besov space. They are associated with Pareto distributions and highly differ from the priors of strong Besov spaces that are Gaussian. Using simulations of these distributions, we build visual representations of the 'typical enemies'. Finally, we exploit these distributions to build a minimax thresholding estimation procedure, called ParetoThresh, that we study from a practical point of view. Subsequently, we consider the heteroscedastic white noise model and under the maxiset approach, we prove that linear estimators are outperformed by adaptive thresholding ones. Finally, we investigate the best way to modelize the sparsity of a sequence throughout a Bayesian approach. For this purpose, we study the maxisets of the classical estimators - median, mean - associated with a model built on heavy tailed densities. The maximal spaces for these rules are Lorentz spaces, and coincide with maxisets associated with thresholding estimators. This result is reinforced by a necessary and sufficient condition on the parameters of the model in order to make sure that the prior is almost surely concentrated on a precise Lorentz space.
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https://tel.archives-ouvertes.fr/tel-00002149
Contributor : Vincent Rivoirard <>
Submitted on : Thursday, December 19, 2002 - 5:59:22 PM
Last modification on : Thursday, November 12, 2020 - 3:10:13 PM
Long-term archiving on: : Friday, April 2, 2010 - 8:03:02 PM

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  • HAL Id : tel-00002149, version 1

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Vincent Rivoirard. Estimation bayésienne non paramétrique. Mathématiques [math]. Université Paris-Diderot - Paris VII, 2002. Français. ⟨tel-00002149⟩

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