Doctor of Philosophy / Department of Statistics / Paul Nelson / Panels of judges are often used to estimate consumer preferences for m items such as food products. Judges can either evaluate each item on several ordinal scales and indirectly produce an overall ranking, or directly report a ranking of the items. A complete ranking orders all the items from best to worst. A partial ranking, as we use the term, only reports rankings of the best q out of m items. Direct ranking, the subject of this report, does not require the widespread but questionable practice of treating ordinal measurement as though they were on ratio or interval scales. Here, we develop and study segmentation models in which the panel may consist of relatively homogeneous subgroups, the segments. Judges within a subgroup will tend to agree among themselves and differ from judges in the other subgroups. We develop and study the statistical analysis of mixture models where it is not known to which segment a judge belongs or in some cases how many segments there are. Viewing segment membership indicator variables as latent data, an E-M algorithm was used to find the maximum likelihood estimators of the parameters specifying a mixture of Mallow’s (1957) distance models for complete and partial rankings. A simulation study was conducted to evaluate the behavior of the E-M algorithm in terms of such issues as the fraction of data sets for which the algorithm fails to converge and the sensitivity of initial values to the convergence rate and the performance of the maximum likelihood estimators in terms of bias and mean square error, where applicable. A Bayesian approach was developed and credible set estimators was constructed. Simulation was used to evaluate the performance of these credible sets as confidence sets. A method for predicting segment membership from covariates measured on a judge was derived using a logistic model applied to a mixture of Mallows probability distance models. The effects of covariates on segment membership were assessed. Likelihood sets for parameters specifying mixtures of Mallows distance models were constructed and explored.

Simple random sampling (SRS) is the most commonly used sampling design in data collection. In many applications (e.g., in fisheries and medical research) quantification of the variable of interest is either time-consuming or expensive but ranking a number of sampling units, without actual measurement on them, can be done relatively easy and at low cost. In these situations, one may use rank-based sampling (RBS) designs to obtain more representative samples from the underlying population and improve the efficiency of the statistical inference. In this thesis, we study the theory and application of the finite mixture models (FMMs) under RBS designs. In Chapter 2, we study the problems of Maximum Likelihood (ML) estimation and classification in a general class of FMMs under different ranked set sampling (RSS) designs. In Chapter 3, deriving Fisher information (FI) content of different RSS data structures including complete and incomplete RSS data, we show that the FI contained in each variation of the RSS data about different features of FMMs is larger than the FI contained in their SRS counterparts. There are situations where it is difficult to rank all the sampling units in a set with high confidence. Forcing rankers to assign unique ranks to the units (as RSS) can lead to substantial ranking error and consequently to poor statistical inference. We hence focus on the partially rank-ordered set (PROS) sampling design, which is aimed at reducing the ranking error and the burden on rankers by allowing them to declare ties (partially ordered subsets) among the sampling units. Studying the information and uncertainty structures of the PROS data in a general class of distributions, in Chapter 4, we show the superiority of the PROS design in data analysis over RSS and SRS schemes. In Chapter 5, we also investigate the ML estimation and classification problems of FMMs under the PROS design. Finally, we apply our results to estimate the age structure of a short-lived fish species based on the length frequency data, using SRS, RSS and PROS designs.