Transformations to Reduce Boundary Bias in Kernel Density Estimation

Abstract

SUMMARY: We consider kernel estimation of a univariate density whose support is a compact interval. If the density is non-zero at either boundary, then the usual kernel estimator can be seriously biased. ‘Reflection’ at a boundary removes some bias, but unless the first derivative of the density is 0 at the boundary the estimator with reflection can still be much more severely biased at the boundary than in the interior. We propose to transform the data to a density that has its first derivative equal to 0 at both boundaries. The density of the transformed data is estimated, and an estimate of the density of the original data is obtained by a change of variables. The transformation is selected from a parametric family, which is allowed to be quite general in our theoretical study. We propose algorithms where the transformation is either a quartic polynomial, a β cumulative density function (CDF) or a linear combination of a polynomial and a β CDF. The last two types of transformation are designed to accommodate possible poles at the boundaries. The first two algorithms are tested on simulated data and compared with an adjusted kernel method of Rice. We find that our proposal performs similarly to Rice's for densities with one-sided derivatives at the boundaries. Unlike Rice's method, our proposal is guaranteed to produce non-negative estimates. This can be a distinct advantage when the density is 0 at either boundary. Our algorithm for densities with poles outperforms the Rice adjustment when the density has a pole at a boundary.