On the asymptotic distribution theory of a class of consistent estimators of a distribution satisfying a uniform stochastic ordering constraint

Abstract

We identify the asymptotic behavior of the estimators proposed by Rojo and Samaniego and Mukerjee of a distribution $F$ assumed to be uniformly stochastically smaller than a known baseline distribution $G$.We show that these estimators are $\sqrt{n}$-convergent to a limit distribution with mean squared error smaller than or equal to the mean squared error of the empirical survival function. By examining the mean squared error of the limit distribution, we are able to identify the optimal estimator within Mukerjee’s class under a variety of different assumptions on $F$ and $G$. Similar theoretical results are developed for the two-sample problem where$F$ and $G$ are both unknown. The asymptotic distribution theory is applied to obtain confidence bands for the survival function $\bar{F}$ based on published data from an accelerated life testing experiment.