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

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.

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.

DOI : 10.1214/12-AIHP516
Classification : 62G05
Mots clés : curve estimation, model selection, composite functions, adaptation, single index model, artificial neural networks, gaussian mixtures
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     title = {Estimating composite functions by model selection},
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Baraud, Yannick; Birgé, Lucien. Estimating composite functions by model selection. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 1, pp. 285-314. doi : 10.1214/12-AIHP516. http://www.numdam.org/articles/10.1214/12-AIHP516/

[1] N. Akakpo. Adaptation to anisotropy and inhomogeneity via dyadic piecewise polynomial selection. Math. Methods Statist. 21 (2012) 1-28. | MR

[2] Y. Baraud. Estimator selection with respect to Hellinger-type risks. Probab. Theory Related Fields 151 (2011) 353-401. | MR

[3] Y. Baraud, F. Comte and G. Viennet. Model selection for (auto-)regression with dependent data. ESAIM Probab. Stat. 5 (2001) 33-49. | Numdam | MR | Zbl

[4] Y. Baraud, C. Giraud and S. Huet. Gaussian model selection with an unknown variance. Ann. Statist. 37 (2009) 630-672. | MR | Zbl

[5] A. R. Barron, L. Birgé and P. Massart. Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 (1999) 301-413. | MR | Zbl

[6] A. R. Barron. Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inform. Theory 39 (1993) 930-945. | MR | Zbl

[7] A. R. Barron. Approximation and estimation bounds for artificial neural networks. Machine Learning 14 (1994) 115-133. | Zbl

[8] L. Birgé. Model selection via testing: An alternative to (penalized) maximum likelihood estimators. Ann. Inst. Henri Poincaré Probab. Stat. 42 (2006) 273-325. | Numdam | MR

[9] L. Birgé. Model selection for Poisson processes. In Asymptotics: Particles, Processes and Inverse Problems, Festschrift for Piet Groeneboom 32-64. E. Cator, G. Jongbloed, C. Kraaikamp, R. Lopuhaä and J. Wellner (Eds). IMS Lecture Notes - Monograph Series 55. Inst. Math. Statist., Beachwood, OH, 2007. | MR | Zbl

[10] L. Birgé. Model selection for density estimation with 𝕃 2 -loss. Probab. Theory Related Fields. To appear. Available at http://arxiv.org/abs/1102.2818. | Zbl

[11] L. Birgé and P. Massart. Gaussian model selection. J. Eur. Math. Soc. (JEMS) 3 (2001) 203-268. | MR | Zbl

[12] W. Dahmen, R. Devore and K. Scherer. Multidimensional spline approximation. SIAM J. Numer. Anal. 17 (1980) 380-402. | MR | Zbl

[13] R. Devore and G. Lorentz. Constructive Approximation. Springer, Berlin, 1993. | MR | Zbl

[14] J. Friedman and J. Tukey. A projection pursuit algorithm for exploratory data analysis. IEEE Trans. Comput. C-23 (1974) 881-890. | Zbl

[15] R. Hochmuth. Wavelet characterizations for anisotropic Besov spaces. Appl. Comput. Harmon. Anal. 12 (2002) 179-208. | MR | Zbl

[16] J. L. Horowitz and E. Mammen. Rate-optimal estimation for a general class of nonparametric regression models with unknown link functions. Ann. Statist. 35 (2007) 2589-2619. | MR | Zbl

[17] P. J. Huber. Projection pursuit (with discussion). Ann. Statist. 13 (1985) 435-525. | MR | Zbl

[18] A. B. Juditsky, O. V. Lepski and A. B. Tsybakov. Nonparametric estimation of composite functions. Ann. Statist. 37 (2009) 1360-1404. | MR | Zbl

[19] C. Maugis and B. Michel. A non asymptotic penalized criterion for Gaussian mixture model selection. ESAIM Probab. Stat. 15 (2011) 41-68. | Numdam | MR

[20] C. J. Stone. Optimal global rates of convergence for nonparametric regression. Ann. Statist. 10 (1982) 1040-1053. | MR | Zbl

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