Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Sharp optimality and some effects of dominating bias in density deconvolution

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 ».