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

CNRS U.M.R. 7599
| ||

``Probabilités et Modèles Aléatoires''
| ||

**Auteur(s): **

**Code(s) de Classification MSC:**

- 62G05 Estimation
- 62G20 Asymptotic properties

**Résumé:** In this paper we consider a kernel estimator of a density in a
convolution model and give a central limit theorem for its
integrated square error (ISE). The kernel estimator is rather
classical in minimax theory when the underlying density is
recovered from noisy observations. The kernel is fixed and depends
heavily on the distribution of the noise, supposed entirely known.
The bandwidth is not fixed, the results hold for any sequence of
bandwidths decreasing to 0. In particular the central limit
theorem holds for the bandwidth minimizing the mean integrated
square error (MISE). Rates of convergence are sensibly different
in the case of regular noise and of super-regular noise. The
smoothness of the underlying unknown density is relevant for the
evaluation of the MISE.

**Mots Clés:** *Convolution density estimation ; Nonparametric density estimation ; Central Limit Theorem ;
Integrated Squared Error ; Noisy observations*

**Date:** 2004-02-02

**Prépublication numéro:** *PMA-880*