Research Article

Exact adaptive pointwise estimation on Sobolev classes of densities

Butucea, Cristina

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.

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 , over the density functions that belong to the Sobolev class W_n(β,L). We consider the adaptive problem setup, where the regularity parameter β is unknown and varies in a given set B _n . A sharp adaptive estimator is obtained, and the explicit asymptotical constant, associated to its rate of convergence is found.

(Received November 15 2000)

(Revised April 9 2001)

(Online publication August 15 2002)

Key Words:

• Density estimation;
• exact asymptotics;
• pointwise risk;
• sharp adaptive estimator.

Mathematics Subject Classification:

• 62N01;
• 62N02;
• 62G20