Penalization versus Goldenshluger − Lepski strategies in warped bases regression

MAP5 UMR CNRS 8145, University Paris Descartes, 45 rue des Saints-Pères, 75006 Paris, France
gaelle.chagny@parisdescartes.fr

Received: 21 June 2011
Revised: 27 October 2011

Abstract

This paper deals with the problem of estimating a regression function f, in a random design framework. We build and study two adaptive estimators based on model selection, applied with warped bases. We start with a collection of finite dimensional linear spaces, spanned by orthonormal bases. Instead of expanding directly the target function f on these bases, we rather consider the expansion of h = f ∘ G^-1, where G is the cumulative distribution function of the design, following Kerkyacharian and Picard [Bernoulli 10 (2004) 1053–1105]. The data-driven selection of the (best) space is done with two strategies: we use both a penalization version of a “warped contrast”, and a model selection device in the spirit of Goldenshluger and Lepski [Ann. Stat. 39 (2011) 1608–1632]. We propose by these methods two functions, ĥ_l (l = 1, 2), easier to compute than least-squares estimators. We establish nonasymptotic mean-squared integrated risk bounds for the resulting estimators, if G is known, or (l = 1,2) otherwise, where Ĝ is the empirical distribution function. We study also adaptive properties, in case the regression function belongs to a Besov or Sobolev space, and compare the theoretical and practical performances of the two selection rules.