Variable window width kernel estimates of probability densities

Modifications of estimators proposed by Breiman, Meisel and Purcell and Abramson, which have variable window widths, are seen to have very fast rates of convergence.

Abstract

Kernel density estimators which allow different amounts of smoothing at different locations are studied. Modifications of estimators proposed by Breiman, Meisel and Purcell (1977) and Abramson (1982a), which have variable window widths, are seen to have very fast rates of convergence. These rates have traditionally been obtained using a less natural higher order kernel, which has the disadvantage of allowing an estimator which takes on negative values.