Inclusion probabilities in partially rank ordered set sampling

In a finite population setting, this paper considers a partially rank ordered set (PROS) sampling design. The PROS design selects a simple random sample (SRS) of M units without replacement from a finite population and creates a partially rank ordered judgment subsets by dividing the units in SRS into subsets of a pre-specified size. The subsetting process creates a partial ordering among units in which each unit in subset h is considered to be smaller than every unit in subset h^' for h^'>h. The PROS design then selects a unit for full measurement from one of these subsets. Remaining units are returned to the population based on three replacement policies. For each replacement policy, we compute the first and second order inclusion probabilities and use them to construct the Horvitz-Thompson estimator and its variance for the estimation of the population total and mean. It is shown that the replacement policy that does not return any of the M units, prior to selection of the next unit for full measurement, outperforms all other replacement policies.