Modifying the kernel distribution function estimator towards reduced bias

Abstract

We explore the convergence rates of a kernel-based distribution function estimator with variable bandwidth. As in density estimation, a considerable bias reduction from O(h 2) to O(h 4) can be obtained by replacing the bandwidth h by h/f 1/2(X i ). We show that the necessary replacement of f 1/2 by some pilot estimator , depending on a second bandwidth g, has no penalizing effect on bias and variance, provided we undersmooth with the pilot bandwidth g, that is g/h→0 in a certain way. Owing to the considerable bias reduction, a simple plug-in normal reference bandwidth selector works effectively in practice. Distribution function estimators with good convergence properties and with simple bandwidth selectors are desirable for repetitive use in smoothed bootstrap algorithms.