Histogram Selection in Non Gaussian Regression

We deal with the problem of choosing a piecewise constant estimator of a regression function s mapping X into 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 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. Received November 26, 2006. Revised November 13, 2007.