Statistics/Probability Theory
1 sparsity and applications in estimation
Comptes Rendus. Mathématique, Volume 344 (2007) no. 6, pp. 399-402.

In this Note, we study the asymptotic behaviour of a new class of penalized M-estimators, built with an 1 type penalty. We prove that adding an 1 constraint enables to construct adaptive estimators, in the sense that the estimators converge at the optimal rate of convergence without prior knowledge of the regularity of the function to be reconstructed. Moreover, we show how the usual issues in nonparametric estimation, such as density estimation, estimation of a regression function and inverse problem estimation can be handled with this methodology.

Nous étudions les propriétés asymptotiques d'une nouvelle classe de M-estimateurs pénalisés par une pénalité de type norme 1. Nous montrons que nous pouvons ainsi construire des estimateurs adaptatifs, c'est-à-dire convergeant à la vitesse optimale sans connaître la régularité de la fonction à estimer. Nous montrons que ce procédé général s'applique dans le cadre du modèle de régression, des problèmes inverses et pour l'estimation de densités.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2007.01.027
Loubes, Jean-Michel 1

1 UMR CNRS 5149, Département de mathématiques et de modélisation, université Montpellier 2, 34095 Montpellier, France
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Loubes, Jean-Michel. $ {\ell }^{1}$ sparsity and applications in estimation. Comptes Rendus. Mathématique, Volume 344 (2007) no. 6, pp. 399-402. doi : 10.1016/j.crma.2007.01.027. http://www.numdam.org/articles/10.1016/j.crma.2007.01.027/

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