Smooth non-parametric estimation of the distribution function from balanced ranked set samples

In certain sampling schemes, such as ranked set sampling, the observed data set can consist entirely of independent order statistics. A balanced ranked set sample consists of n independent cycles, in each of which the experimenter selects k random samples, each of size k. The recorded observation in sample r of cycle i is the rth order statistic, X-(r). Stokes and Sager (1988) have shown that the empirical distribution function based on such samples is an unbiased estimate of the underlying distribution function and is more precise than the empirical distribution function calculated from a simple random sample of the same size. In this article, the properties of the empirical distribution function developed by Stokes and Sager (1988) are further investigated. In addition, a kernel-type estimator of the distribution function is developed. For large n, the smooth estimator is shown to be consistent and asymptotically normal. Finally, small sample properties of the two estimators are investigated and compared via computer simulations. Copyright (C) 2004 John Wiley Sons, Ltd.