On the estimation of smooth densities by strict probability densities at optimal rates in sup-norm

Abstract

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.