Estimating composite functions by model selection
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 1, p. 285-314

We consider the problem of estimating a function s on [-1,1] k for large values of k by looking for some best approximation of s by composite functions of the form gu. Our solution is based on model selection and leads to a very general approach to solve this problem with respect to many different types of functions g,u and statistical frameworks. In particular, we handle the problems of approximating s by additive functions, single and multiple index models, artificial neural networks, mixtures of Gaussian densities (when s is a density) among other examples. We also investigate the situation where s=gu for functions g and u belonging to possibly anisotropic smoothness classes. In this case, our approach leads to a completely adaptive estimator with respect to the regularities of g and u.

Cet article traite du problème de l’estimation d’une fonction s définie sur [-1,1] k lorsque k est grand en utilisant des approximations de s par des fonctions composées de la forme gu. Notre solution est fondée sur la sélection de modèle et conduit, pour résoudre ce problème, à une approche très générale tant sur les possibilités de choix des fonctions g et u que sur les cadres statistiques d’application. En particulier, et entre autres exemples, nous considérons l’approximation de s par des fonctions additives, des modèles de type “single” ou “multiple index”, des réseaux de neurones, ou des mélanges de densités gaussiennes lorsque s est elle-même une densité. Nous étudions également le cas où s est exactement de la forme gu pour des fonctions g et u appartenant à des classes de régularités qui peuvent être anisotropes. Dans ce cas, notre approche conduit à un estimateur complètement adaptatif par rapport aux régularités de g et u.

DOI : https://doi.org/10.1214/12-AIHP516
Classification:  62G05
Keywords: curve estimation, model selection, composite functions, adaptation, single index model, artificial neural networks, gaussian mixtures
@article{AIHPB_2014__50_1_285_0,
     author = {Baraud, Yannick and Birg\'e, Lucien},
     title = {Estimating composite functions by model selection},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {1},
     year = {2014},
     pages = {285-314},
     doi = {10.1214/12-AIHP516},
     zbl = {1281.62093},
     mrnumber = {3161532},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_1_285_0}
}
Baraud, Yannick; Birgé, Lucien. Estimating composite functions by model selection. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 1, pp. 285-314. doi : 10.1214/12-AIHP516. http://www.numdam.org/item/AIHPB_2014__50_1_285_0/

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