Ranked set sampling (RSS) is an alternative to simple random sampling that has been shown to outperform simple random sampling (SRS) in many situations. RSS outperforms SRS by reducing the variance of an estimator, thereby providing the same accuracy with a smaller sample size than is needed in simple random sampling. Ranked set sampling involves the preliminary ranking of potential sample units on the variable of interest using judgment or an auxiliary variable to aid in sample selection. Ranked set sampling prescribes the number of units from each rank order to be measured. Chapter 1 provides an overview of RSS.
Chapter 2 considers unbalanced RSS and allocations associated with it under the assumption that we can observe the measurement of interest on each unit. Balanced ranked set sampling assigns equal numbers of sample units to each rank order. Unbalanced ranked set sampling allows unequal allocation to the various ranks, but this allocation may be sensitive to the quality of the information available to do the allocation. In Chapter 2, we use a simulation study to conduct a sensitivity analysis of optimal allocation of sample units to each of the order statistics in unbalanced ranked set sampling. Our motivating example comes from the National Survey of Families and Households.
In Chapter 3, we consider the optimal allocations for unbalanced RSS when allowing the ranking method to be imperfect. We provide a general formula for the optimal allocation under any imperfect ranking procedure. Then we investigate the optimal allocation scheme for several variations of imperfect rankings and examine graphs that display the changing allocations.
In Chapter 4, we consider missing data in the balanced ranked set sampling (RSS) setting under the condition of perfect rankings. Our goal is to estimate a population proportion. We study estimations of the parameters for RSS data under three missing data models: 1) missing completely at random (MCAR); 2) missing at random (MAR); and 3) non-ignorable non-response (NINR). In the MAR case we allow the probability of missingness to depend on the known RSS ranking. In the NINR case we allow the probability of missingness to depend on whether the observation that is measured is a success or failure. We present the theoretical results for MLEs of the parameters under these three models, followed by an example with data from the National Survey of Families and Households.
Chapter 5 derives the variances associated with the estimators we find in Chapter 4. In providing some measure of the variability of our estimate, we can then use inferential statistics to analyze our estimators. Chapter 6 uses the variances from Chapter 5 to derive optimal allocation schemes for estimators.