Adaptive Density Flattening--A Metric Distortion Principle for Combating Bias in Nearest Neighbor Methods

Abstract

With a wide variety of approaches to density estimation, it is profitable to perturb the data so as to make 2nd order derivatives of their density vanish. An adaptive transformation to local uniformity for instance will (for unchanged variance) lower bias to a vanishing fraction of what a Rosenblatt-Parzen or nearest neighbor estimator on the raw data yields; fractional pilot sampling, a common technical device of little practical appeal, can be shown by an embedding argument to be dispensable. An upshot is that MSE can be lowered by attacking the variance directly through extra smoothing, without the usual penalty from inflated bias.