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:** We consider estimation of the common probability density $f$ of
i.i.d. random variables $X_i$ that are observed with an additive
i.i.d. noise. We assume that the unknown density $f$ belongs to a
class ${\cal A}$ of densities whose characteristic function is
described by the exponent $\exp(-\alpha |u|^r)$ as $|u|\to\infty$,
where $\alpha>0$, $r>0$. The noise density is supposed to be known
and such that its characteristic function decays as $\exp
(-\beta|u|^s)$, as $|u|\to\infty$, where $\beta>0$, $s>0$.
Assuming that $r < s$,, we suggest a kernel type estimator that is optimal in sharp asymptotical minimax sense on ${\cal A}$
simultaneously under the pointwise and the $\mathbb{L}_2$-risks.
The variance of this estimator turns out to be asymptotically
negligible w.r.t. its squared bias. For $r < s/2$ we construct a sharp adaptive estimator of $f$. We discuss some effects of dominating bias, such as superefficiency of minimax estimators.

**Mots Clés:** *Deconvolution ; nonparametric density estimation ; infinitely
differentiable functions ; exact constants in nonparametric
smoothing ; minimax risk ; adaptive curve estimation*

**Date:** 2004-03-31

**Prépublication numéro:** *PMA-898*

**Updated version :** PMA-898Bis.pdf (09/07/2004) Its title has been changed to
** « Sharp optimality for density deconvolution with dominating bias
».**