Histogram selection in non gaussian regression
ESAIM: Probability and Statistics, Tome 13 (2009), pp. 70-86.

We deal with the problem of choosing a piecewise constant estimator of a regression function s mapping 𝒳 into . We consider a non gaussian regression framework with deterministic design points, and we adopt the non asymptotic approach of model selection via penalization developed by Birgé and Massart. Given a collection of partitions of 𝒳, with possibly exponential complexity, and the corresponding collection of piecewise constant estimators, we propose a penalized least squares criterion which selects a partition whose associated estimator performs approximately as well as the best one, in the sense that its quadratic risk is close to the infimum of the risks. The risk bound we provide is non asymptotic.

DOI : 10.1051/ps:2008002
Classification : 62G08, 62G05
Mots clés : CART, change-points detection, deviation inequalities, model selection, oracle inequalities, regression
@article{PS_2009__13__70_0,
     author = {Sauv\'e, Marie},
     title = {Histogram selection in non gaussian regression},
     journal = {ESAIM: Probability and Statistics},
     pages = {70--86},
     publisher = {EDP-Sciences},
     volume = {13},
     year = {2009},
     doi = {10.1051/ps:2008002},
     mrnumber = {2502024},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps:2008002/}
}
TY  - JOUR
AU  - Sauvé, Marie
TI  - Histogram selection in non gaussian regression
JO  - ESAIM: Probability and Statistics
PY  - 2009
SP  - 70
EP  - 86
VL  - 13
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps:2008002/
DO  - 10.1051/ps:2008002
LA  - en
ID  - PS_2009__13__70_0
ER  - 
%0 Journal Article
%A Sauvé, Marie
%T Histogram selection in non gaussian regression
%J ESAIM: Probability and Statistics
%D 2009
%P 70-86
%V 13
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps:2008002/
%R 10.1051/ps:2008002
%G en
%F PS_2009__13__70_0
Sauvé, Marie. Histogram selection in non gaussian regression. ESAIM: Probability and Statistics, Tome 13 (2009), pp. 70-86. doi : 10.1051/ps:2008002. http://archive.numdam.org/articles/10.1051/ps:2008002/

[1] Y. Baraud, Model selection for regression on a fixed design. Probab. Theory Related Fields 117 (2000) 467-493. | MR | Zbl

[2] Y. Baraud, F. Comte and G. Viennet, Model Selection for (auto-)regression with dependent data. ESAIM: PS 5 (2001) 33-49. | EuDML | Numdam | MR | Zbl

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

[4] L. Birgé and P. Massart, Minimal penalties for gaussian model selection. To be published in Probab. Theory Related Fields (2005). | MR | Zbl

[5] L. Birgé and Y. Rozenholc, How many bins should be put in a regular histogram. ESAIM: PS 10 (2006) 24-45. | EuDML | Numdam | MR | Zbl

[6] O. Bousquet, Concentration Inequalities for Sub-Additive Functions Using the Entropy Method. Stochastic Inequalities and Applications 56 (2003) 213-247. | MR | Zbl

[7] L. Breiman, J. Friedman, R. Olshen and C. Stone, Classification And Regression Trees. Chapman et Hall (1984). | MR | Zbl

[8] G. Castellan, Modified Akaike’s criterion for histogram density estimation. C.R. Acad. Sci. Paris Sér. I Math. 330 (2000) 729-732. | MR | Zbl

[9] O. Catoni, Universal aggregation rules with sharp oracle inequalities. Ann. Stat. (1999) 1-37.

[10] E. Lebarbier, Quelques approches pour la détection de ruptures à horizon fini. Ph.D. thesis, Université Paris XI Orsay (2002).

[11] G. Lugosi and A. Nobel, Consistency of data-driven histogram methods for density estimation and classification. Ann. Stat. 24 (1996) 786-706. | MR | Zbl

[12] C.L. Mallows, Some comments on c p . Technometrics 15 (1973) 661-675. | Zbl

[13] P. Massart, Notes de Saint-Flour. Lecture Notes to be published (2003).

[14] A. Nobel, Histogram regression estimation using data-dependent partitions. Ann. Stat. 24 (1996) 1084-1105. | MR | Zbl

[15] M. Sauvé, Sélection de modèles en régression non gaussienne. Applications à la sélection de variables et aux tests de survie accélérés. Ph.D. thesis, Université Paris XI Orsay (2006).

[16] M. Sauvé and C. Tuleau, Variable selection through CART. Research Report 5912, INRIA (2006).

Cité par Sources :