A Review of Bootstrap Methods in Ranked Set Sampling

Introduction

The sampling design is a fundamental part of any statistical inference. A well-developed sampling design plays a critical role in ensuring that data can capture the distinct characteristics of the population and are sufficient to draw the conclusions needed. Choice of a sampling plan depends on many aspects, including but not limited to the purpose of the analysis, the rarity of the characteristics under study, the nature of the total population, the size of the area to be studied, and most importantly, the cost of the study. One of the most commonly cited sampling plans is the simple random sampling (SRS). In practice, a more structured sampling mechanism, such as stratified sampling or systematic sampling, may be obtained to achieve a representative sample of the population of interest. Ranked-set sampling (RSS) is an alternative data collection method and has been known as a cost-efficient sampling procedure for many years. This approach to data collection was first proposed by McIntyre (1952) to improve the precision of estimated pasture yield. Later, Takahasi and Wakimoto (1968) established a rigorous statistical foundation for the theory of RSS. The ranked set sampling (RSS) utilizes the basic intuitive properties associated with simple random sampling (SRS). However, it involves the extra structure induced through the judgment ranking and the independence of the resulting order statistics. As a result, the procedures based on RSS lead to more efficient estimators of population parameters than those based on an SRS with the same sample size. The existing literature also includes works on hypothesis testing and point and interval estimation under both parametric and nonparametric settings. The most basic version of RSS is the balanced RSS. The process of generating an RSS involves drawing k² units at random from the target population. These items are then randomly divided into k sets of k units each. Within each set, the units are then ranked by some means other than a direct measurement. For example, visually or using a concomitant measurement that is comparatively cheaper to measure and easy to obtain than the measurement of interest. Finally, one item from each set is chosen for actual quantification. To be more specific, from the first set, we select the item with the smallest judgment-rank for measurement, from the second set, we choose the item with the second smallest judgment-rank, and so on, until the unit ranked largest is selected from the k^th set. This complete procedure, called a cycle, is repeated independently m times to obtain a ranked set sample of size mk. Therefore, a balanced RSS of size mk requires a total of mk² units to be selected, but only mk of them are measured. Hence, a more comprehensive range of the population can be covered while significantly reducing the sampling cost. According to Takahasi and Wakimoto (1968), for easy implementation of RSS, the set size k is usually kept as four or less. However, we can obtain a large sample by increasing the cycle size m. Another option is that of unbalanced RSS. In an unbalanced RSS, n×k units are selected at random from the target population. These items are then randomly divided into n sets of k units each. Units in each set are judgment ranked without measuring the actual units. In this setting, let m_r denote the number of sets allocated to measure units having the r^th judgment-rank such that . The measured observations then constitute an unbalanced RSS of size n.