Statistica Sinica 15(2005), 73-98

UNIMODAL KERNEL DENSITY ESTIMATION

BY DATA SHARPENING

Peter Hall$^{1}$ and Kee-Hoon Kang$^{1,2}$

$^1$Australian National University and $^2$Hankuk University of Foreign Studies

Abstract: We discuss a robust data sharpening method for rendering a standard kernel estimator, with a given bandwidth, unimodal. It has theoretical and numerical properties of the type that one would like such a technique to enjoy. In particular, we show theoretically that, with probability converging to 1 as sample size diverges, our technique alters the kernel estimator only in places where the latter has spurious bumps, and is identical to the kernel estimator in places where that estimator is monotone in the correct direction. Moreover, it automatically splices together, in a smooth and seamless way, those parts of the estimator that it leaves unchanged and those that it adjusts. Provided the true density is unimodal our estimator generally reduces mean integrated squared error of the standard kernel estimator.

Key words and phrases: Bandwidth, data sharpening, heavy tailed distributions, kernel methods, mean squared error, nonparametric density estimation, order constraint, tilting methods, unimodal density.