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

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

Minimax estimation of the noise level and of the signal density in a semiparametric convolution model


Code(s) de Classification MSC:

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