| 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:
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 ».