A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the Degree of Doctor of Philosophy. Johannesburg, August 2016. / There is a growing demand for small area estimates for policy and decision making, local planning and fund distribution. Surveys are generally designed to give representative estimates at national or regional level, but estimates of variables of interest are often also needed at the small area levels. These cannot be reliably obtained from the survey data as the sample sizes at these levels are too small. This problem is addressed by using small area estimation techniques. The main aim of this thesis is to develop confidence intervals (CIs) which are accurate to terms O(m–3/2 ) under the FH model using the Taylor series expansion. Rao (2003a), among others, notes that there is a situation in mixed model estimation that the estimates of the variance component of the random effect, A, can take negative values. In this case, Prasad and Rao (1990) consider ˆA = 0. Under this situation, the contribution of the mean squared error (MSE) estimate, assuming all parameters are known, becomes zero. As a solution, Rao (2003a) among others proposed a weighted estimator with fixed weights (i.e., wi = 12 ). In addition, if the MSE estimate is negative, we cannot construct CIs based on the empirical best linear unbiased predictor (EBLUP) estimates. Datta, Kubokawa, Molina and Rao (2011) derived the MSE estimator for the weighted estimator with fixed weights which is always positive. We use their MSE estimator to derive CIs based on this estimator to overcome the above difficulties. The other criticism of the MSE estimator is that it is not area-specific since it does not involve the direct estimator in its expression. Following Rao (2001), we propose area specific MSE estimators and use them to construct CIs. The performance of the proposed CIs are investigated via simulation studies and compared with the Cox (1975) and Prasad and Rao (1990) methods. Our simulation results show that the proposed CIs have higher coverage probabilities. These methods are applied to standard poverty and percentage of food expenditure measures estimated from the 2010/11 Household Consumption Expenditure survey and the 2007 census data sets. Keywords: Small area estimation, Weighted estimator with fixed weights, EBLUP, FH model, MSE, CI, Poverty, percentage of food expenditure / LG2017

A Monte Carlo simulation technique was employed in this study to determine if the r transformation, a test of homogeneity of variance, affords adequate protection against Type I error over a range of equal sample sizes and number of groups when samples are obtained from normal and non-normal distributions. Additionally, this study sought to determine if the r transformation is more robust than Bartlett's chi-square to deviations from normality. Four populations were generated representing normal, uniform, symmetric leptokurtic, and skewed leptokurtic distributions. For each sample size (6, 12, 24, 48), number of groups (3, 4, 5, 7), and population distribution condition, the r transformation and Bartlett's chi-square were calculated. This procedure was replicated 1,000 times; the actual significance level was determined and compared to the nominal significance level of .05. On the basis of the analysis of the generated data, the following conclusions are drawn. First, the r transformation is generally robust to violations of normality when the size of the samples tested is twelve or larger. Second, in the instances where a significant difference occurred between the actual and nominal significance levels, the r transformation produced (a) conservative Type I error rates if the kurtosis of the parent population were 1.414 or less and (b) an inflated Type I error rate when the index of kurtosis was three. Third, the r transformation should not be used if sample size is smaller than twelve. Fourth, the r transformation is more robust in all instances to non-normality, but the Bartlett test is superior in controlling Type I error when samples are from a population with a normal distribution. In light of these conclusions, the r transformation may be used as a general utility test of homogeneity of variances when either the distribution of the parent population is unknown or is known to have a non-normal distribution, and the size of the equal samples is at least twelve.

Clustering is the process of finding underlying group structure in data. As the scale of data collection continues to grow, this “big data” phenomenon results in more complex data structures. These data structures are not always compatible with traditional clustering methods, making their use problematic. This thesis presents methodology for analyzing samples of four-way and higher data, examples of these more complex data types. These data structures consist of samples of continuous data arranged in multidimensional arrays. A large emphasis is placed on clustering this data using mixture models that leverage tensor-variate distributions to model the data. Parameter estimation for all these methods are based on the expectation-maximization algorithm. Both simulated and real data are used for illustration. / Thesis / Doctor of Science (PhD)

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The primary objective of this study was to quantitatively analyze compressive static stress distributions across pallet deck surfaces supporting flexible and rigid packaging in simulated warehouse storage systems. Three different densities of polyolefin foams (2, 4, and 6 lb/ft3, pcf) simulated a variety of flexible and rigid packaging with a range of stiffness properties. A layer of single wall C-flute corrugated fiberboard acted as a sensing medium and also simulated the bottom of a corrugated box. Pressure sensitive films were used to detect compressive static stresses at the interface between the polyolefin foams and the pallet deckboard. Image analysis computer software program was developed to quantitatively characterize stress distributions left on pressure sensitive film. 280 lbs of compression load were applied to a Plexiglas® pallet section (40 x 3.5 inches, L x W) with ¾ inch deck thickness, as well as to a steel pallet section (40 x 3.5 inches, L x W) with ½ inch deck thickness. In both cases, the pallet sections were used in a simulated pallet storage rack. 700 lbs of compression load were applied to the same steel pallet section that was used in the racking simulation and the Plexiglas® pallet sections (40 x 3.5 inches, L x W) with ½ and ¾ inch deck thicknesses were used in simulated block (floor) stack storage to measure the stress distributions and deflections of deckboards. Applying the final models of resultant non-uniform stress distributions enabled the development of finite element analysis (FEA) models of pallet deckboard deflections. The predicted FEA models of the deckboard deflections were validated through comparison with experimentally measured deflections in the simulated warehouse storage systems. In the final models, the resultant three foams' stress distributions across pallet deck surfaces in both rack and floor stack storage simulations were non-uniform. The changes in the degree of stress concentrations and maximum stress levels along the deckboards varied, depending on the stiffness of the foams and deckboards and the support conditions in the simulated warehouse storage models. Qualified test indicates that the 2pcf and 4pcf foams represent non-rigid sack products and the 6pcf foam represents rigid packaging and contents. All tests were conducted within a few minutes; hence, all test data were assumed to be initially resulted compressive stresses. The compressive stresses may change over time. The measure of stress concentrations is the stress intensity factor, which is the ratio of initial maximum resultant compressive stress to the applied stress. The initial maximum resultant compressive stresses were adjusted for rate of loading which varied due to the difference in the stiffness of the foams. The table below shows the adjusted initial maximum resultant compressive stress intensity factors. The product of the calculation uniformly distributed compressive stress and the stress intensity factor is the appropriate criteria for designing packaging of product with adequate compressive strength. These factors will be useful when designing pallets, packaging, and unit loads.In simulated block stack storage, the foam stiffness (package and product stiffness) had a more significant effect on the stress distributions and concentrations along the deckboards than did the pallet deck stiffness. As a result, the stiffer foam presented a greater change in stress levels along the deckboard under the compression load. The quantified and evaluated stress concentrations and stress distributions will be useful in understanding the interactions between pallets and packaging, reducing product damage and improving the safety of the work place during the long-term storage of the unit loads. The predicted FEA models will allow the industry to better optimize pallets, packaging, and unit load designs. / Master of Science

<p> This thesis deals with metrizations of sets of conditional entropies and sets of sub-σ-algebras. G Co Rajski's Theorem ([9]) on the metric space of discrete probability distributions can be deduced as a particular case of a theorem on the metric space of sub-σ-algebras given in Chapter III, the proof of which is comparatively very concise. The completeness of this metric space and some other properties are also proved. </p> / Thesis / Master of Science (MSc)

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