Ranked set sampling (RSS) is an efficient data collection framework compared to simple random sampling (SRS). It is widely used in various application areas such as agriculture, environment, sociology, and medicine, especially in situations where measurement is expensive but ranking is less costly. Most past research in RSS focused on situations where the underlying distribution is continuous. However, it is not unusual to have a discrete data generation mechanism. Estimating statistical functionals are challenging as ties may truly exist in discrete RSS. In this thesis, we started with estimating the cumulative distribution function (CDF) in discrete RSS. We proposed two methods to incorporate the information brought by ties. The first method is based on the idea of Frey (2012), which only works for the balanced RSS. The second one is based on the NPMLE method proposed by Kvam and Samaniego (1994). The second method can be applied in both balanced and unbalanced RSS. By simulation studies, we showed that the new methods improve the efficiency of estimation. Later, we proposed the corresponding plug-in estimators for the population mean and the population variance. The new estimators showed higher efficiency compared to the existing estimators in the literature.
Another problem considered in this thesis is to improve the estimation efficiency of each order stratum CDF when tie information is not available. We proposed a new estimator by imposing uniformly stochastic ordering constraint on the order strata CDF's. By using the "ranking" relationship between the order strata CDF's, the new estimator showed a higher efficiency for the strata except the edge strata (the smallest and largest order stratum).
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Cui, Heng, "Discrete Ranked Set Sampling" (2018). Statistical Science Theses and Dissertations. 2.