Optimal model selection in density estimation
Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 3, p. 884-908

In order to calibrate a penalization procedure for model selection, the statistician has to choose a shape for the penalty and a leading constant. In this paper, we study, for the marginal density estimation problem, the resampling penalties as general estimators of the shape of an ideal penalty. We prove that the selected estimator satisfies sharp oracle inequalities without remainder terms under a few assumptions on the marginal density s and the collection of models. We also study the slope heuristic, which yields a data-driven choice of the leading constant in front of the penalty when the complexity of the models is well-chosen.

Une procédure de pénalisation en sélection de modèle repose sur la construction d’une forme pour la pénalité ainsi que sur le choix d’une constante de calibration. Dans cet article, nous étudions, pour le problème d’estimation de la densité, les pénalités obtenues par rééchantillonnage de pénalités idéales. Nous montrons l’efficacité de ces procédures pour l’estimation de la forme des pénalités en prouvant, pour les estimateurs sélectionnés, des inégalités d’oracle fines sans termes résiduelles; les résultats sont valides sous des hypothèses faibles à la fois sur la densité inconnue s et sur les collections de modèles. Ces pénalités sont de plus faciles à calibrer puisque la constante asymptotiquement optimale peut être calculée en fonction des poids de rééchantillonnage. En pratique, le nombre de données est toujours fini, nous étudions donc également l’heuristique de pente et justifions l’algorithme de pente qui permet de calibrer la constante de calibration à partir des données.

DOI : https://doi.org/10.1214/11-AIHP425
Classification:  62G07,  62G09
Keywords: density estimation, optimal model selection, resampling methods, slope heuristic
@article{AIHPB_2012__48_3_884_0,
     author = {Lerasle, Matthieu},
     title = {Optimal model selection in density estimation},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {48},
     number = {3},
     year = {2012},
     pages = {884-908},
     doi = {10.1214/11-AIHP425},
     zbl = {1244.62052},
     mrnumber = {2976568},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2012__48_3_884_0}
}
Lerasle, Matthieu. Optimal model selection in density estimation. Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 3, pp. 884-908. doi : 10.1214/11-AIHP425. http://www.numdam.org/item/AIHPB_2012__48_3_884_0/

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