Statistical inference under a stochastic ordering constraint in ranked set sampling

Abstract

This paper introduces estimators for the judgment class and population cumulative distribution functions (CDF) under stochastic order restriction. The estimators are defined as the minimizer of a version of the Cramér-von Mises distance function. It is shown that the new estimators are strongly and uniformly consistent for the judgment class population distributions and have smaller integrated mean square errors than the integrated mean square errors of the empirical CDF estimators. The proposed estimators are used to calibrate the effect of imperfect ranking on statistical procedures. It is shown that this calibration works quite well in ranked set sample Mann–Whitney–Wilcoxon rank-sum and sign tests. The use of estimators and calibration procedure are illustrated on a ranked set sample data.