We consider a posteriori error estimators that can be applied to anisotropic tetrahedral finite element meshes, i.e. meshes where the aspect ratio of the elements can be arbitrarily large. Two kinds of Zienkiewicz-Zhu (ZZ) type error estimators are derived which are both based on some recovered gradient. Two different, rigorous analytical approaches yield the equivalence of both ZZ error estimators to a known residual error estimator. Thus reliability and efficiency of the ZZ error estimation is obtained. Particular attention is paid to the requirements on the anisotropic mesh. The analysis is complemented and confirmed by several numerical examples.

This thesis is focused on local polynomial smoothers of the conditional vari- ance function in a heteroscedastic nonparametric regression model. Both mean and variance functions are assumed to be smooth, but neither is assumed to be in a parametric family. The basic idea is to apply a local linear regression to squa- red residuals. This method, as we have shown, has high minimax efficiency and it is fully adaptive to the unknown conditional mean function. However, the local linear estimator may give negative values in finite samples which makes variance estimation impossible. Hence Xu and Phillips proposed a new variance estimator that is asymptotically equivalent to the local linear estimator for interior points but is guaranteed to be non-negative. We also established asymptotic results of both estimators for boundary points and proved better asymptotic behavior of the local linear estimator. That motivated us to propose a modification of the local li- near estimator that guarantees non-negativity. Finally, simulations are conducted to evaluate the finite sample performances of the mentioned estimators.

Binomial group testing has been long recognized as an efficient method of estimating proportion of subjects with a specific characteristic. The method is superior to the classic maximum likelihood estimator (MLE), particularly when the proportion is small. Under the group testing model, we assume the testing is conducted without error. In the present research, a new Bayes estimator will be proposed that utilizes an additional piece of information, the proportion to be estimated is small and within a given range. It is observed that with the appropriate choice of the hyper-parameter our new Bayes estimator has smaller mean squared error (MSE) than the classic MLE, Burrows estimator, and the existing Bayes estimator. Furthermore, on the basis of heavy Monte Carlo simulation we have determined the best hyper-parameters in the sense that the corresponding new Bayes estimator has the smallest MSE. A table of these best hyper-parameters is made for proportions within the considered range.

Multiple linear regression model plays a key role in statistical inference and it has extensive applications in business, environmental, physical and social sciences. Multicollinearity has been a considerable problem in multiple regression analysis. When the regressor variables are multicollinear, it becomes difficult to make precise statistical inferences about the regression coefficients. There are some statistical methods that can be used, which are discussed in this thesis are ridge regression, Liu, two parameter biased and LASSO estimators. Firstly, an analytical comparison on the basis of risk was made among ridge, Liu and LASSO estimators under orthonormal regression model. I found that LASSO dominates least squares, ridge and Liu estimators over a significant portion of the parameter space for large dimension. Secondly, a simulation study was conducted to compare performance of ridge, Liu and two parameter biased estimator by their mean squared error criterion. I found that two parameter biased estimator performs better than its corresponding ridge regression estimator. Overall, Liu estimator performs better than both ridge and two parameter biased estimator.

In this thesis different situations are considered in which the preliminary test estimator is applied and the performance of the preliminary test estimator under different proposed loss functions, namely the reflected normal , linear exponential (LINEX) and bounded LINEX (BLINEX) loss functions is evaluated. In order to motivate the use of the BLINEX loss function rather than the reflected normal loss or the LINEX loss function, the risk for the preliminary test estimator and its component estimators derived under BLINEX loss is compared to the risk of the preliminary test estimator and its components estimators derived under both reflected normal loss and LINEX loss analytically (in some sections) and computationally. It is shown that both the risk under reflected normal loss and the risk under LINEX loss is higher than the risk under BLINEX loss. The key focus point under consideration is the estimation of the regression coefficients of a multiple regression model under two conditions, namely the presence of multicollinearity and linear restrictions imposed on the regression coefficients. In order to address the multicollinearity problem, the regression coefficients were adjusted by making use of Hoerl and Kennard’s (1970) approach in ridge regression. Furthermore, in situations where under- or overestimation exist, symmetric loss functions will not give optimal results and it was necessary to consider asymmetric loss functions. In the economic application, it was shown that a loss function which is both asymmetric and bounded to ensure a maximum upper bound for the loss, is the most appropriate function to use. In order to evaluate the effect that different ridge parameters have on the estimation, the risk values were calculated for all three ridge regression estimators under different conditions, namely an increase in variance, an increase in the level of multicollinearity, an increase in the number of parameters to be estimated in the regression model and an increase in the sample size. These results were compared to each other and summarised for all the proposed estimators and proposed loss functions. The comparison of the three proposed ridge regression estimators under all the proposed loss functions was also summarised for an increase in the sample size and an increase in variance. / Thesis (PhD)--University of Pretoria, 2014. / lk2014 / Statistics / PhD / Unrestricted

In this thesis, we discuss calibration estimation in the presence of nonresponse with a focus on the linear calibration estimator and the propensity calibration estimator, along with the use of different levels of auxiliary information, that is, sample and population levels. This is a fourpapers- based thesis, two of which discuss estimation in two steps. The two-step-type estimator here suggested is an improved compromise of both the linear calibration and the propensity calibration estimators mentioned above. Assuming that the functional form of the response model is known, it is estimated in the first step using calibration approach. In the second step the linear calibration estimator is constructed replacing the design weights by products of these with the inverse of the estimated response probabilities in the first step. The first step of estimation uses sample level of auxiliary information and we demonstrate that this results in more efficient estimated response probabilities than using population-level as earlier suggested. The variance expression for the two-step estimator is derived and an estimator of this is suggested. Two other papers address the use of auxiliary variables in estimation. One of which introduces the use of principal components theory in the calibration for nonresponse adjustment and suggests a selection of components using a theory of canonical correlation. Principal components are used as a mean to accounting the problem of estimation in presence of large sets of candidate auxiliary variables. In addition to the use of auxiliary variables, the last paper also discusses the use of explicit models representing the true response behavior. Usually simple models such as logistic, probit, linear or log-linear are used for this purpose. However, given a possible complexity on the structure of the true response probability, it may raise a question whether these simple models are effective. We use an example of telephone-based survey data collection process and demonstrate that the logistic model is generally not appropriate.

Age-Period-Cohort models offer a useful framework to study trends of time-specific phenomena in various areas. Yet the perfect linear relationship among age, period, and cohort induces a singular design matrix and brings about the identification issue of age, period, and cohort model due to the identity Cohort = Period -- Age. Over the last few decades, multiple methods have been proposed to cope with the identification issue, e.g., the intrinsic estimator (IE), which may be viewed as a limiting form of ridge regression. This study views the ridge estimator from a Bayesian perspective by introducing a prior distribution(s) for the ridge parameter(s). Data used in this study describe the incidence rate of cervical cancer among Ontario women from 1960 to 1994. Results indicate that a Bayesian ridge model with a common prior for the ridge parameter yields estimates of age, period, and cohort effects similar to those based on the intrinsic estimator and to those based on a ridge estimator. The performance of Bayesian models with distinctive priors for the ridge parameters of age, period, and cohort effects is affected more by the choice of prior distributions. In sum, a Bayesian ridge model is an alternative way to deal with the identification problem of age, period, and cohort model. Future studies should further investigate the influences of different prior choices on Bayesian ridge models. / text

The main focus of many agricultural, ecological and environmental studies is to develop well designed, cost-effective and efficient sampling designs. Ranked set sampling (RSS) is one of those sampling methods that can help accomplish such objectives by incorporating prior information and expert knowledge to the design. In this thesis, new RSS schemes are suggested for efficiently estimating the population mean. These sampling schemes can be used as cost-effective alternatives to the traditional simple random sampling (SRS) and RSS schemes. It is shown that the mean estimators under the proposed sampling schemes are at least as efficient as the mean estimator with SRS. We consider the best linear unbiased estimators (BLUEs) and the best linear invariant estimators (BLIEs) for the unknown parameters (location and scale) of a location-scale family of distributions under double RSS (DRSS) scheme. The BLUEs and BLIEs with DRSS are more precise than their counterparts based on SRS and RSS schemes. We also consider the BLUEs based on DRSS and ordered DRSS (ODRSS) schemes for the unknown parameters of a simple linear regression model using replicated observations. It turns out that, in terms of relative efficiencies, the BLUEs under ODRSS are better than the BLUEs with SRS, RSS, ordered RSS (ORSS) and DRSS schemes. Quality control charts are widely recognized for their potential to be a powerful process monitoring tool of the statistical process control. These control charts are frequently used in many industrial and service organizations to monitor in-control and out-of-control performances of a production or manufacturing process. The RSS schemes have had considerable attention in the construction of quality control charts. We propose new exponentially weighted moving average (EWMA) control charts for monitoring the process mean and the process dispersion based on the BLUEs obtained under ORSS and ODRSS schemes. We also suggest an improved maximum EWMA control chart for simultaneously monitoring the process mean and dispersion based on the BLUEs with ORSS scheme. The proposed EWMA control charts perform substantially better than their counterparts based on SRS and RSS schemes. Finally, some new EWMA charts are also suggested for monitoring the process dispersion using the best linear unbiased absolute estimators of the scale parameter under SRS and RSS schemes.

The Path Flow Estimator (PEE) concept was originally developed to estimate path flows (hence origin-destination flows) and link flows for a whole road network (given some counts at selected roads). It is now further developed as an alternative for modeling different transportation planning applications: (1) a bicycle network analysis tool for non-motorized transportation planning, (2) a multi-class traffic assignment model for freight planning, and (3) a simplified travel demand forecasting framework for small community planning. The first application of the redeveloped PFE is to develop a two-stage bicycle traffic assignment model for estimating/predicting bicycle volumes on a transportation network. The first stage considers key criteria (e.g., distance related attributes, safety related attributes, air quality related attributes etc.) to generate a set of non-dominated (or efficient) paths, while the second stage adopts several traffic assignment methods to determine the flow allocations to the network. This two-stage approach can be used as a stand-alone bicycle traffic assignment to the transportation network given a bicycle origin-destination (O-D) matrix. The second application aims to enhance the realism of traffic assignment models for freight planning by incorporating different modeling considerations into the multi-class traffic assignment problem. These modeling considerations involve developing both model formulation and customized solution algorithm, which in turn involve asymmetric interactions among different vehicle types (i.e., cars versus trucks), a path-size logit (PSL) model (for accounting random perceptions of network conditions with explicit consideration of route overlapping), and various traffic restrictions imposed either individually or together to multiple vehicle types in a transportation network. In the third application, a simplified planning framework is developed to perform planning applications in small communities where limited planning resources hinder the development and application of a full four-step model. Two versions (i.e., base year and future year) of the PFE are proposed to address the specific transportation planning issues and needs of small communities. These new PFE developments for planning applications are tested with different realistic transportation networks. The results suggest that the new PFE applications proposed in this dissertation provide an alternative to the traditional four-step travel demand forecasting model that can be used as a stand-alone application with better modeling capability and fewer resources.

The main purpose of this study was to construct different vertical scales based on various combinations of calibration methods and proficiency estimators to investigate the impact different choices may have on these properties of the vertical scales that result: grade-to-grade growth, grade-to-grade variability, and the separation of grade distributions. Calibration methods investigated were concurrent calibration, separate calibration, and fixed a, b, and c item parameters for common items with simple prior updates (FSPU). Proficiency estimators investigated were Maximum Likelihood Estimator (MLE) with pattern scores, Expected A Posteriori (EAP) with pattern scores, pseudo-MLE with summed scores, pseudo-EAP with summed scores, and Quadrature Distribution (QD). The study used datasets from the Iowa Tests of Basic Skills (ITBS) in the Vocabulary, Reading Comprehension (RC), Math Problem Solving and Data Interpretation (MPD), and Science tests for grades 3 through 8. For each of the research questions, the following conclusions were drawn from the study. With respect to the comparisons of three calibration methods, for the RC and Science tests, concurrent calibration, compared to FSPU and separate calibration, showed less growth and more slowly decreasing growth in the lower grades, less decrease in variability over grades, and less separation in the lower grades in terms of horizontal distances. For the Vocabulary and MPD tests, differences in both grade-to-grade growth and in the separation of grade distributions were trivial. With respect to the comparisons of five proficiency estimators, for all content areas, the trend of pseudo-MLE ≥ MLE > QD > EAP ≥ pseudo-EAP was found in within-grade SDs, and the trend of pseudo-EAP ≥ EAP > QD > MLE ≥ pseudo-MLE was found in the effect sizes. However, the degree of decrease in variability over grades was similar across proficiency estimators. With respect to the comparisons of the four content areas, for the Vocabulary and MPD tests compared to the RC and Science tests, growth was less, but somewhat steady, and the decrease in variability over grades was less. For separation of grade distributions, it was found that the large growth suggested by larger mean differences for the RC and Science tests was reduced through the use of effect sizes to standardize the differences.