Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Adaptive boxcar deconvolution on full Lebesgue measure sets

Auteur(s):

Code(s) de Classification MSC:

• 62G05 Estimation
• 62G08 Nonparametric regression

Résumé: We consider the nonparametric estimation of a function that is observed in white noise after convolution with a boxcar, the indicator of an interval $(-a,a)$. In a recent paper \citet{jkpr04} have developped a wavelet deconvolution algorithm (called {\tt WaveD}) that can be used for certain'' boxcar kernels. For example, {\tt WaveD} can be tuned to achieve near optimal rates over Besov spaces when $a$ is a Badly Approximable (BA) irrational number. While the set of all BA's contains quadratic irrationals e.g. $a=\sqrt{5}$ it has Lebesgue measure zero, however. In this paper we derive two tuning scenarios of {\tt WaveD} that are valid for almost all'' boxcar convolution (i.e. when $a\in A$ where $A$ is a full Lebesgue measure set). We propose (i) a tuning inspired from Minimax theory over Besov spaces; (ii) a tuning inspired from Maxiset theory providing similar rates as for BA numbers. Asymptotic theory informs that (i) in the worst case scenario, departure from the BA assumption, affects {\tt WaveD} convergence rates, at most, by log factors; (ii) the Maxiset tuning, which yields smaller thresholds, is superior to the Minimax (conservative) tuning over a whole range of Besov sup-scales. Our asymptotic results are illustrated in an extensive simulation of boxcar convolution observed in white noise.

Mots Clés: Adaptive estimation ; deconvolution ; non-parametric regression ; Meyer wavelet

Date: 2004-09-24

Prépublication numéro: PMA-934