Volume 17, 2013
|Page(s)||328 - 358|
|Published online||17 May 2013|
Penalization versus Goldenshluger − Lepski strategies in warped bases regression
MAP5 UMR CNRS 8145, University Paris Descartes,
45 rue des Saints-Pères,
Revised: 27 October 2011
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, ĥl (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.
Mathematics Subject Classification: 62G05 / 62G08
Key words: Adaptive estimator / model selection / nonparametric regression estimation / warped bases
© EDP Sciences, SMAI, 2013
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.