Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**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*