Boundary Kernels for Distribution Function Estimation

Abstract

Boundary effects for kernel estimators of curves with compact supports are well known in regression and density estimation frameworks. In this paper we address the use of boundary kernels for distribution function estimation. We establish the ChungSmirnov law of iterated logarithm and an asymptotic expansion for the mean integrated squared error of the proposed estimator. These results show the superior theoretical performance of the boundary modified kernel estimator over the classical kernel estimator for distribution functions that are not smooth at the extreme points of the distribution support. The automatic selection of the bandwidth is also briefly discussed in this paper. Beta reference distribution and cross-validation bandwidth selectors are considered. Simulations suggest that the cross-validation bandwidth performs well, although the simpler reference distribution bandwidth is quite effective for the generality of test distributions.