Uniform asymptotics for kernel density estimators with variable bandwidths

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’, Annals of Statistics, 23, 1–10] modification of Abramson's [Abramson, I. (1982), ‘On Bandwidth Variation in Kernel Estimates – A Square-root Law’, 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.