Variable window width kernel density estimators, with the width varying proportionally to the square root of the density, have been thought to have superior asymptotic properties. The rate of convergence has been claimed to be as good as those typical for higher-order kernels, which makes the variable width estimators more attractive because no adjustment is needed to handle the negativity usually entailed by the latter. However, in a recent paper, Terrell and Scott show that these results can fail in important cases. In this paper, we characterize situations where the fast rate is valid, and also give rates for a variety of cases where they are slower. In addition, a modification of the usual variable window width estimator is proposed, which does have the earlier claimed rates of convergence.
"Improved Variable Window Kernel Estimates of Probability Densities." Ann. Statist. 23 (1) 1 - 10, February, 1995. https://doi.org/10.1214/aos/1176324451