In this dissertation, the problem of partitioning a set of treatments with respect to a control treatment is considered. Starting in 1950's a number of researchers have worked on this problem and have proposed alternative solutions. In Tong (1979), the authors proposed a formulation to solve this problem and hundreds of researchers and practitioners have used that formulation for the partition problem. However, Tong's formulation is somewhat rigid and misleading for the practitioners, if the distance between the ``good'' and the ``bad'' populations is large. In this case, the indifference zone gets quite large and the undesirable feature of the Tong's formulation to partition the populations in the indifference zone, without any penalty, can potentially lead Tong's formulation to produce misleading partitions. In this dissertation, a generalization of the Tong's formulation is proposed, under which, the treatments in the indifference zone are not partitioned as ``good'' or ``bad'', but are partitioned as a identifiable set. For this generalized partition, a fully sequential and a two-stage procedure is proposed and its theoretical properties are derived. The proposed procedures are also studied via Monte Carlo Simulation studies. The thesis concludes with some non-parametric partition procedures and the study of robustness of the various available procedures in the statistical literature.

It is a standard practice in small area estimation (SAE) to use a model-based approach to borrow information from neighboring areas or from areas with similar characteristics. However, survey data tend to have gaps, ties and outliers, and parametric models may be problematic because statistical inference is sensitive to parametric assumptions. We propose nonparametric hierarchical Bayesian models for multi-stage finite population sampling to robustify the inference and allow for heterogeneity, outliers, skewness, etc. Bayesian predictive inference for SAE is studied by embedding a parametric model in a nonparametric model. The Dirichlet process (DP) has attractive properties such as clustering that permits borrowing information. We exemplify by considering in detail two-stage and three-stage hierarchical Bayesian models with DPs at various stages. The computational difficulties of the predictive inference when the population size is much larger than the sample size can be overcome by the stick-breaking algorithm and approximate methods. Moreover, the model comparison is conducted by computing log pseudo marginal likelihood and Bayes factors. We illustrate the methodology using body mass index (BMI) data from the National Health and Nutrition Examination Survey and simulated data. We conclude that a nonparametric model should be used unless there is a strong belief in the specific parametric form of a model.

Even a modest positive autocorrelation results in a considerable increase in the number of false alarms that are produced when applying a CUSUM chart. Knowledge of the process to be controlled allows for suitable adaptation of the CUSUM procedure. If one has to suspect the normality assumption, nonparametric control procedures such as the rank-based CUSUM chart are a practical alternative. The paper reports the results of a simulation study on the robustness (in terms of sensitivity of the ARL) of the rank-based CUSUM chart against serial correlation of the control variable. The results indicate that the rank-based CUSUM chart is less affected by correlation than the observation-based chart: The rank-based CUSUM chart shows a smaller increase in the number of false alarms and a higher decrease in the ARL in the out-of-control case than the the observation-based chart. (author's abstract) / Series: Forschungsberichte / Institut für Statistik