Semiparametric deconvolution with unknown noise variance

Matias, Catherine

ESAIM: Probability and Statistics, Tome 6 (2002), pp. 271-292.

Ce travail porte sur l’estimation semi-paramétrique dans un modèle de convolution, où le bruit suit une loi gaussienne centrée de variance ${\sigma }^2$ inconnue. Les modèles non-paramétriques de convolution, où la loi du bruit est entièrement connue, ont été largement étudiés, et il s’avère que la vitesse d’estimation de la densité $g$ du signal est d’autant plus lente que la loi du bruit est régulière [3]. Cependant, la régularité imposée sur $g$ permet d’améliorer ces vitesses d’estimation [15]. Dans cet article, nous montrons que lorsque la loi du bruit (qui est supposée gaussienne centrée) possède une variance ${\sigma }^2$ inconnue (ce qui en pratique est toujours le cas), les vitesses d’estimation de la densité $g$ du signal sont dégradées par rapport au cas où la loi du bruit est entièrement connue. Plus précisemment, se restreindre à des densités régulières pour le signal n’améliore pas la vitesse d’estimation de $g$ qui est toujours plus lente que ${(logn)}^{-1}$. Nous construisons alors deux estimateurs de la variance, dont un consistant dès que le signal a un moment d’ordre $1$ fini. Nous mentionnons enfin une conséquence de ce travail qui est la détérioration des vitesses d’estimation des paramètres dans un modèle de régression non-linéaire avec erreurs sur les variables.

This paper deals with semiparametric convolution models, where the noise sequence has a gaussian centered distribution, with unknown variance. Non-parametric convolution models are concerned with the case of an entirely known distribution for the noise sequence, and they have been widely studied in the past decade. The main property of those models is the following one: the more regular the distribution of the noise is, the worst the rate of convergence for the estimation of the signal’s density $g$ is [3]. Nevertheless, regularity assumptions on the signal density $g$ improve those rates of convergence [15]. In this paper, we show that when the noise (assumed to be gaussian centered) has a variance ${\sigma }^2$ that is unknown (actually, it is always the case in practical applications), the rates of convergence for the estimation of $g$ are seriously deteriorated, whatever its regularity is supposed to be. More precisely, the minimax risk for the pointwise estimation of $g$ over a class of regular densities is lower bounded by a constant over $logn$. We construct two estimators of ${\sigma }^2$, and more particularly, an estimator which is consistent as soon as the signal has a finite first order moment. We also mention as a consequence the deterioration of the rate of convergence in the estimation of the parameters in the nonlinear errors-in-variables model.

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