Uniform asymptotics for kernel density estimators with variable bandwidths

TL;DR

This paper establishes the optimal asymptotic properties of a modified variable bandwidth kernel density estimator, ensuring accurate density estimation with certain smoothness and boundedness conditions.

Contribution

It proves that the Hall, Hu, and Marron modification of Abramson's estimator achieves optimal asymptotic behavior under specified smoothness and boundedness assumptions.

Findings

Estimator satisfies optimal asymptotic properties

Effective for densities with four continuous derivatives

Works uniformly on bounded sets where the preliminary estimate is bounded away from zero

Abstract

It is shown that the Hall, Hu and Marron [Hall, P., Hu, T., and Marron J.S. (1995), Improved Variable Window Kernel Estimates of Probability Densities, {\it Annals of Statistics}, 23, 1--10] modification of Abramson's [Abramson, I. (1982), On Bandwidth Variation in Kernel Estimates - A Square-root Law, {\it Annals of Statistics}, 10, 1217--1223] variable bandwidth kernel density estimator satisfies the optimal asymptotic properties for estimating densities with four uniformly continuous derivatives, uniformly on bounded sets where the preliminary estimator of the density is bounded away from zero.