Smooth simultaneous confidence bands for cumulative distribution functions

Abstract

A plug-in kernel estimator is proposed for Hölder continuous cumulative distribution function (cdf) based on a random sample. Uniform closeness between the proposed estimator and the empirical cdf estimator is established, while the proposed estimator is smooth instead of a step function. A smooth simultaneous confidence band is constructed based on the smooth distribution estimator and the Kolmogorov distribution. Extensive simulation study using two different automatic bandwidths confirms the theoretical findings.