Open Access
VOL. 9 | 2013 On the estimation of smooth densities by strict probability densities at optimal rates in sup-norm
Evarist Giné, Hailin Sang

Editor(s) M. Banerjee, F. Bunea, J. Huang, V. Koltchinskii, M. H. Maathuis

Inst. Math. Stat. (IMS) Collect., 2013: 128-149 (2013) DOI: 10.1214/12-IMSCOLL910


It is shown that the variable bandwidth density estimators proposed by McKay [ Canad. J. Statist. 21 (1993) 367–375; Variable kernel methods in density estimation (1993) Queen’s University] following earlier findings by Abramson [ Ann. Statist. 10 (1982) 1217–1223] approximate density functions in $C^4(\mathbb{R}^d)$ at the minimax rate in the supremum norm over bounded sets where the preliminary density estimates on which they are based are bounded away from zero. A somewhat more complicated estimator proposed by Jones, McKay and Hu [ Ann. Inst. Statist. Math. (1994) 46 521–535] to approximate densities in $C^6(\mathbb{R})$ can also be shown to attain minimax rates in sup norm over the same kind of sets. These estimators are strict probability densities.


Published: 1 January 2013
First available in Project Euclid: 8 March 2013

zbMATH: 1325.62082
MathSciNet: MR3186753

Digital Object Identifier: 10.1214/12-IMSCOLL910

Primary: 62G07

Keywords: clipping filter , kernel density estimator , rates of convergence , Spatial adaptation , square root law , sup-norm loss , variable bandwidth

Rights: Copyright © 2010, Institute of Mathematical Statistics

Back to Top