Uniform Convergence of an Estimator of a Distribution Function

A class of estimators of a distribution function, which includes the empirical distribution function, is discussed. The necessary and sufficient condition for the estimator to be asymptotically unbiased at all continuity points of the distribution function is presented. We give the asymptotic evaluation of the variance and show its asymptotic normality. The necessary and sufficient condition for the estimator to converge uniformly to a continuous distribution function with probability one is presented (the continuity can be delected in case of the necessary condition). We propose an estimator of a p -th quantile based on the estimator of a distribution function, which converges to the p -th quantile with probability one under certain conditions.