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Exact adaptive pointwise estimation on Sobolev classes of densities

Published online by Cambridge University Press:  15 August 2002

Cristina Butucea
Cristina Butucea*
Affiliation:
Université Paris 10, Modal'X, bâtiment G, 200 avenue de la République, 92001 Nanterre, France; cbutucea@u-paris10.fr. and Université Paris 6, Laboratoire Probabilités et Modèles Aléatoires, 6 rue Clisson, 75013 Paris, France; butucea@ccr.jussieu.fr.
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Abstract

The subject of this paper is to estimate adaptively the common probability density of n independent, identically distributed random variables. The estimation is done at a fixed point $x_{0}\in \mathbb R$, over the density functions that belong to the Sobolev class Wn(β,L). We consider the adaptive problem setup, where the regularity parameter β is unknown and varies in a given set Bn. A sharp adaptive estimator is obtained, and the explicit asymptotical constant, associated to its rate of convergence is found.

Keywords

Density estimationexact asymptoticspointwise risksharp adaptive estimator.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 5 , 2001 , pp. 1 - 31
Copyright
© EDP Sciences, SMAI, 2001

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