Bootstrapping Empirical Distributions under Auxiliary Information

Abstract

Being the nonparametric maximum likelihood estimator, the classical empirical distribution function is the estimator of choice for a completely unknown distribution function. As shown by Qin and Lawless (1994), in the presence of some auxiliary information the nonparametric maximum likelihood estimator is a modified empirical distribution function. It puts random masses on the observations in order to take the available information into account Zhang (1997) has proved a functional central limit theorem for this modified empirical distribution function. The centered Gaussian limit process in this fclt has a complicated covariance structure so that the result is not directly applicable in statistical problems, e.g. for the construction of confidence bands. It is shown here that the bootstrap is one possible remedy.