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 (f) over cap (1/2)(g), depending on a secondbandwidth g, has nopenalizing effect onbias andvariance, 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.