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:**

- 62G07 Density estimation
- 62G20 Asymptotic properties
- 62C20 Minimax procedures

**Résumé:** We consider a semiparametric convolution model of an unknown signal
with supersmooth noise having unknown scale parameter.
We construct a consistent estimation procedure for the noise level and
prove that its rate is optimal in the minimax sense.
For identifiability reasons, the noise has to be smoother than the signal
in this problem. Two convergence rates
are distinguished according to different smoothness properties for the
signal. In one case the rate is sharp optimal, i.e. the asymptotic value of
the risk is evaluated up to a constant. Moreover, we construct a consistent
estimator of the signal, by using a plug-in method in the classical kernel
estimation procedure. We establish that the
estimation of the signal is deteriorated comparatively to the case of
entirely known noise distribution. In fact, nonparametric rates of convergence
are governed by the rate of estimation of the noise level. We also prove that
those rates are minimax (or nearly minimax in a few specific cases).
Simulation results bring
new ideas on practical use of our estimation algorithms.

**Mots Clés:** *Analytic densities ; deconvolution ; minimax estimation ; noise level ;
pointwise risk ; semiparametric model ; Sobolev classes*

**Date:** 2003-02-26

**Prépublication numéro:** *PMA-795*