Mean Estimation Based on FWA Using Ranked Set Sampling with Single and Multiple Rankers
The Ranked Set Sampling (RSS) is an advanced sampling method which improves the precision and accuracy of the mean estimator. In RSS, the units in the random sets which are drawn from a population are ranked by a ranking mechanism, and one of these ranked
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Department of Statistics, Dokuz Eylul University, 35170 Izmir, Turkey {bekir.cetintav,selma.erdogan,neslihan.ortabas}@deu.edu.tr Department of Industrial Engineering, Izmir Univerity, 35170 Izmir, Turkey [email protected]
Abstract. The Ranked Set Sampling (RSS) is an advanced sampling method which improves the precision and accuracy of the mean estimator. In RSS, the units in the random sets which are drawn from a population are ranked by a ranking mechanism, and one of these ranked units is sampled from each set with a specific scheme. Ranking the units (visually or by a concomitant variable) could not be perfect because there is an uncertainty in decision making about the rank of a unit. In this study, we propose a fuzzy set perspective for RSS and an estimator for the population mean based on Fuzzy Weighted Average (FWA) operator. A real data application is given to illustrate the new approach for the single and multiple rankers. Keywords: Ranked set sampling · Uncertainty weighted average · Multiple rankers
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Introduction
Ranked set sampling (RSS) is a useful and alternative sampling method where the knowledge about the ranks (orders) of the units is used. Because of the additional information on the ranks of the units, RSS is more representative than the simple random sampling (SRS) counterpart with equal sample size. Takahasi and Wakimoto [1], and Dell and Clutter [2] construct the statistical background of this method, introduced earlier by McIntyre [3], and show that the efficiency of the mean estimator based on RSS is greater than or equal to the efficiency of the mean estimator based on SRS. For detailed information about RSS, see the book of Chen et al. [4] and the review of Wolfe [5]. In simple terms, RSS procedure consists of three parts. Random sets are drawn from a specific population, B. Cetintav—This study is supported by the Scientific and Technological Research Council of Turkey (TUBITAK-COST Grant No. 115F300) under ISCH COST Action IS1304. c Springer International Publishing Switzerland 2016 J.P. Carvalho et al. (Eds.): IPMU 2016, Part II, CCIS 611, pp. 790–797, 2016. DOI: 10.1007/978-3-319-40581-0 64
Mean Estimation Based on FWA with Ranked Set Sampling
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the units in the sets are ranked by a mechanism, and one of these ranked units is sampled from each set with a specific scheme. In the ranking mechanism, the ranker could be an expert-researcher or a highly-correlated concomitant variable and makes the decisions about the rank of the units. In practice, the decisions of ranks should be made without actual measurement of the concerning variable. Thus, these decisions could not be always perfect even if ranking is done by a powerful criterion. This unavoidable uncertainty in the ranking mechanism is mentioned as imprecise/imperfect ranking in the literature. Several studies, Bohn and Wolfe [6], MacEachern et al. [7], Frey [8], Oztrk [9–11] focused on the modeling uncertainty in a probabilistic way. According to us, fuzzy sets could be an alternative way
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