Cluster sampling is a survey design that is commonly used when a simple random sample may be too costly or inefficient to implement for a population. The idea of a cluster sample is that a population can be divided into groups called clusters. The usual cluster sample consists of sampling n of the N clusters within a population. Within these clusters, a further sample may be taken of secondary sampling units. When a further sample is taken within the primary sample, this is called a two-stage cluster sampling design.
Design-based estimates of mean and total are well-established for two-stage cluster samples. For certain populations, however, a Bayesian approach may be preferred for ease of interpretation or to estimate a parameter not well-developed in the design-based literature. When a Bayesian approach is used, a commonly desired property of the model is that the sampled and unsampled units are exchangeable. In this way, the data and not the model is influencing population estimates.
A Bayesian approach in which the sampled and unsampled units are exchangeable is the Bayes Urn model. In this dissertation, we will show how the Bayes Urn model can provide admissible estimators under any finite parameter space. Using simulation studies for a variety of two-stage cluster samples, we will show how the Bayes Urn estimates are similar to the standard design-based estimates. As an extension, we will use the Bayes Urn model to incorporate auxiliary information for two-stage cluster samples. For both of these cases we will compare the Bayes Urn estimate of the population total to the standard design-based estimates as well as two standard Bayesian estimates. In the case when auxiliary information is available for a two-stage cluster sample, there is no well-developed Bayesian approach in the literature. In addition to population total, we will also consider estimates of cluster median, in which no design-based approach is well-developed in the literature. Through these simulation studies, the admissibility proof, and the flexible properties inherent of the Bayes Urn model, we will see that this approach can provide useful estimations in complex designs beyond the two-stage cluster sample design.