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

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

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

- 62G07 Density estimation
- 60J60 Diffusion processes [See also 58J65]
- 62C20 Minimax procedures
- 62G20 Asymptotic properties

**Résumé:** We derive sharp asymptotic minimax bounds (that is, bounds which
concern the exact asymptotic constant of the risk) for
nonparametric density estimation based on discretely observed
diffusion processes. We study two particular problems for which
there already exist such results in the case of independent and
identically distributed observations, namely, minimax density
estimation in Sobolev classes with $L_{2}$-loss and in H\"{o}lder
classes with $L_\infty$-loss.
We derive independently lower and upper bounds for the asymptotic minimax
risks and show that they coincide with the classical efficiency bounds. It
is proven that these bounds can be attained by usual kernel density
estimators. The lower bounds are obtained by analyzing the problem of
estimating the marginal density in certain families of processes, $\left\{
\{X_i^f\}, f\in{\cal F}_n \right\}$, which are shrinking neighborhoods of
around some central process, $\{X_i^{f_0}\}$, in the sense that the
set of densities ${\cal F}_n$ forms a shrinking neighborhood centered around
$f_0$.

**Mots Clés:** *Density estimation ; dependent data ; diffusion processes ; discrete
sampling ; exact asymptotics ; minimax risk ; nonparametric estimation *

**Date:** 2003-06-24

**Prépublication numéro:** *PMA-832*