Sampling method plays an important role for data collection in a scientific research. Ranked set sampling (RSS), which was first introduced by McIntyre, is an advanced method to obtain data for getting information and inference about the population of interest. The main impact of RSS is to use the ranking information of the units in the sampling mechanism. Even though most of theoretical inferences are made based on exact measurement of the variable of interest, the ranking process is done with an expert judgment or concomitant variable (without exact measurement) in practice. Because of the ambiguity in discriminating the rank of one unit with another, ranking the units could not be perfect, and it may cause uncertainty. There are some studies focused on the modeling of this uncertainty with a probabilistic perspective in the literature. In this chapter, another perspective, a fuzzy-set-inspired approach, for the uncertainty in the ranking mechanism of RSS is introduced.
TopIntroduction
Ranked set sampling (RSS) is first introduced by McIntyre (1952) in a biometric research. In simple terms, random sets are drawn from a population, the units in the sets are ranked with a ranking mechanism, and one of these ranked units is sampled from each set with a specific scheme in RSS procedure. The ranking mechanism can be the visual inspection of a human expert or a highly-correlated concomitant variable. When the actual measurement of the variable of interest is difficult or expensive than ranking the experimental units, RSS can be introduced as an easier and cheaper way to obtain a more representative sample. The main impact of RSS is to use the ranking information of units in the sampling mechanism. When the ranking is done properly, the inference based on RSS generally gives effective results comparing with simple random sampling (SRS) for both parametric and non-parametric cases.
The ranking mechanism is one of the major parts of RSS procedure. Ranking the units could not be perfect in practice because there is an ambiguity in discriminating the rank of one unit with another without actual measurement. Clearly, there is an uncertainty in the ranking mechanism of RSS in the cases of imperfect ranking. In the literature, several studies focused on the case of imperfect ranking. The main idea of these studies is to model the uncertainty in ranking mechanism and to search the impact of the imperfect ranking on inference. Dell and Clutter (1972) introduce a pioneering model for imperfect ranking with using an additive model of concerning and concomitant variable. Bohn and Wolfe (1994) propose expected spacings model for the probabilities of imperfect judgment rankings based on the expected differences of units in the set. They suggest constructing a single stochastic matrix consisting of specifying probabilities of units which are inversely proportional to the expected differences between the order statistics. Presnell and Bohn (1999) use this model to show that the RSS procedure is asymptotically at least as efficient as the SRS, even if the ranking is imperfect. Ozturk (2008) proposes inference techniques for RSS data in the case of imperfect ranking using the models of Bohn and Wolfe (1994) and Frey (2007). On the other hand, there are RSS inspired studies where new sampling procedures are defined using the main idea of ranking in RSS, but also the uncertainty in ranking mechanism is studied. MacEachern et al. (2004) introduced Judgment Post-Stratified (JPS) sampling method for a specific case where a simple random sample is already chosen. In JPS method, a random set is taken for each unit in the simple random sample and the units are ranked. Then the rank information of the units is used as an additional information for inference. Ozturk (2011a,b) introduce partially ranked ordered set (PROS) sampling design for a specific case of imperfect ranking. Rankers are allowed to declare any two or more units are tied in ranks whenever the units cannot be ranked with high confidence. In that study, ties are the cause of imprecise rank decisions in the sampling design and the tied units divide the probability equally among the assigned ranks.
In this chapter, we propose a fuzzy set perspective to dealing with the uncertainty occurring in the ranking process of RSS. The most important advantage of using fuzzy sets is to allow the units to belong to different sets with different membership degrees. When we use fuzzy sets, the units in the random sets could belong to not only the most possibly rank but also the other possible ranks. For this purpose, preliminaries for RSS and fuzzy set theory will be given at first. In the following section, fuzzy set inspired approach for modeling of the uncertainty in the ranking process will be introduced. Fuzzy inspired Ranked Set Sampling (FRSS) procedure for single ranker case is also given in this section. An extension for the FRSS procedure is also given to combine the information coming from multiple rankers. The estimator for the population mean based on the FRSS method is given, and some asymptotic properties are discussed. FRSS is also compared with simple random sampling (SRS), RSS and Multiple RSS via Monte Carlo simulation study. An application and a numerical study based on a real data set are conducted.