Model selection for density estimation with L2-loss

Abstract : We consider here estimation of an unknown probability density $s$ belonging to $\Bbb{L}_2(\mu)$ where $\mu$ is a probability measure. We have at hand $n$ i.i.d.\ observations with density $s$ and use the squared $\Bbb{L}_2$-norm as our loss function. The purpose of this paper is to provide an abstract but completely general method for estimating $s$ by model selection, allowing to handle arbitrary families of finite-dimensional (possibly non-linear) models and any $s\in\Bbb{L}_2(\mu)$. We shall, in particular, consider the cases of unbounded densities and bounded densities with unknown $\Bbb{L}_\infty$-norm and investigate how the $\Bbb{L}_\infty$-norm of $s$ may influence the risk. We shall also provide applications to adaptive estimation and aggregation of preliminary estimators. Although of a purely theoretical nature, our method leads to results that cannot presently be reached by more concrete ones.
Type de document :
Pré-publication, Document de travail
32 pages. 2008
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https://hal.archives-ouvertes.fr/hal-00347691
Contributeur : Lucien Birgé <>
Soumis le : mardi 16 décembre 2008 - 15:17:30
Dernière modification le : mercredi 12 octobre 2016 - 01:02:53
Document(s) archivé(s) le : mardi 8 juin 2010 - 17:25:05

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  • HAL Id : hal-00347691, version 1

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Lucien Birgé. Model selection for density estimation with L2-loss. 32 pages. 2008. <hal-00347691>

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