Characterization of a Ranked-Set Sample with Application to Estimating Distribution Functions

Abstract

Ranked-set sampling has been shown to provide improved estimators of the mean and variance when actual measurement of the observations is difficult but ranking of the elements in a sample is relatively easy. This result holds even if the ranking is imperfect. In this article, we provide a characterization of a ranked-set sample that makes the source of this additional information intuitively clear. It is applied to show that the empirical distribution function of a ranked-set sample is unbiased and has greater precision than that from a random sample. The null distribution of a Kolmogorov-Smirnov statistic based on this empirical distribution function is derived for the case in which perfect ranking is possible. It is seen to be stochastically smaller than the usual Kolmogorov-Smirnov statistic based on a simple random sample, resulting in a smaller simultaneous confidence interval for the cumulative distribution function.