Histogram selection in non Gaussian regression

Issue		ESAIM: PS Volume 13, 2009


Page(s)		70 - 86
DOI		10.1051/ps:2008002
Published online		26 March 2009

ESAIM: PS, March 2009, Vol. 13, p. 70-86
DOI: 10.1051/ps:2008002

Histogram selection in non Gaussian regression

Marie Sauvé

Laboratoire de mathématiques – Bâtiment 425, Université Paris Sud, 91405 Orsay Cedex, France; marie.sauve@math.u-psud.f

Received November 26, 2006. Revised November 13, 2007. Published online 26 March 2009

Abstract
We deal with the problem of choosing a piecewise constant estimator of a regression function s mapping $\mathcal{X}$ into $\mathbb{R}$ . 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 $\mathcal{X}$ , 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.

Mathematics Subject Classification. 62G08, 62G05

Key words: CART, change-points detection, deviation inequalities, model selection, oracle inequalities, regression

© EDP Sciences, SMAI 2009

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