On Empirical Distribution Functions 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 in (Ann Statist 22:300–325, 1994), in the presence of some nonparametric auxiliary information about the underlying distribution function the nonparametric maximum likelihood estimator is a modified empirical distribution function. It puts random masses on the observations in order to take the auxiliary information into account. Under a second moment condition Zhang in (Metrika 46:221–244, 1997) has proved a functional central limit theorem for this modified empirical distribution function. The covariance function of the centered Gaussian limit process in his result is smaller than the covariance function of the Browinan brigde limit process in Donsker’s functional central limit theorem for the classical empirical distribution function. If the auxiliary information about the underlying distribution function is knowledge of the mean, then the second moment condition in Zhang’s result requires square integrable random variables. In this note we will study integrable random variables with known mean which are not square integrable and will show that Zhang’s result is no longer true.