IN KERNEL DENSITY ESTIMATIONl

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 o 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 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 beta CDF, or a linear combination of a polynomial and a beta CDF. The last two types of transformations 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 nonnegative 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 does have a pole at a boundary. Many methods for estimating a probability density Ix have been proposed. When Ix is compactly supported, most estimators, including the popular kernel method, are more biased near the endpoints than in the interior. In practical settings, the boundary region is often 20% to 50%, and sometimes more, of the support of lx, so that boundary bias can be a serious problem. In this article, we propose a data transformation method for reducing the boundary bias of kernel estimators. Let Xl' ... , X n be a random sample from Ix. Suppose for convenience that the support of Ix is [0,1]. The conventional kernel estimate of Ix (x) is-1 ~ {x-Xi} Ix (X) = nh ~ K h ' 1=1 where K is the kernel function, and h is a window width depending on n. Here K is assumed to be a symmetric probability density, and to simplify the discussion, we …