Document ESAIM: P&S, August 2007, Vol. 11, pp. 344-364
DOI: 10.1051/ps:2007023
On pointwise adaptive curve estimation based on inhomogeneous data
Stéphane GaïffasLaboratoire de Probabilités et Modèles Aléatoires, U.M.R. CNRS 7599 and Université Paris 7, 175 rue du Chevaleret, 75013 Paris, France; gaiffas@math.jussieu.fr
(Received February 6, 2006. Revised June 20 and October 10, 2006. Published online 17 August 2007.)
Abstract
We want to recover a signal based on noisy inhomogeneous data (the
amount of data can vary strongly on the estimation domain). We model
the data using nonparametric regression with random design, and we
focus on the estimation of the regression at a fixed point x0
with little, or much data. We propose a method which adapts both to
the local amount of data (the design density is unknown) and to the
local smoothness of the regression function. The procedure consists
of a local polynomial estimator with a Lepski type data-driven
bandwidth selector, see for instance
Lepski et al. [Ann. Statist. 25 (1997) 929-947]. We assess this procedure in the
minimax setup, over a class of function with local smoothness s >
0 of Hölder type. We quantify the amount of data at x0 in
terms of a local property on the design density called regular
variation, which allows situations with strong variations in the
concentration of the observations. Moreover, the optimality of the
procedure is proved within this framework.
Mathematics Subject Classification. 62G05, 62G08
Key words: Adaptive estimation, inhomogeneous data, nonparametric regression, random design.
© EDP Sciences, SMAI 2007