IMPROVED VARIABLE WINDOW KERNEL ESTIMATES OF PROBABILITY DENSITIES

Variable window width kernel density estimators, with the width varyin g proportionally to the square root of the density, have been thought to have superior asymptotic properties. The rate of convergence has be en claimed to be as good as those typical for higher-order kernels, wh ich makes the variable width estimators more attractive because no adj ustment is needed to handle the negativity usually entailed by the lat ter. However, in a recent paper, Terrell and Scott show that these res ults can fail in important cases. In this paper, we characterize situa tions 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 usu al variable window width estimator is proposed, which does have the ea rlier claimed rates of convergence.