Ordinal coefficient alpha is a newly derived non-parametric reliability estimate. As with any point estimate, ordinal coefficient alpha is merely an estimate of a population parameter and tends to vary from sample to sample. Researchers report the confidence interval to provide readers with the amount of precision obtained. Several methods with differing computational approaches exist for confidence interval estimation for alpha, including the Fisher, Feldt, Bonner, and Hakstian and Whalen (HW) techniques. Overall, coverage rates for the various methods were unacceptably low with the Fisher method as the highest performer at 62%. Because of the poor performance across all four confidence interval methods, a need exists to develop a method which works well for ordinal coefficient alpha.

Master of Science / Department of Statistics / Paul I. Nelson / Capture-Recaputre (CR) experiments stemmed from the study of wildlife and are widely used in areas such as ecology, epidemiology, evaluation of census undercounts, and software testing, to estimate population size, survival rate, and other population parameters. The basic idea of the design is to use “overlapping” information contained in multiple samples from the population. In this report, we focus on the simplest form of Capture-Recapture experiments, namely, a two-sample Capture-Recapture design, which is conventionally called the “Petersen Method.” We study and compare the performance of three methods of constructing confidence intervals for the population size based on a Capture-Recapture design, asymptotic normality estimation, Chapman estimation, and “inverting a chi-square test” estimation, in terms of coverage rate and mean interval width. Simulation studies are carried out and analyzed using R and SAS. It turns out that the “inverting a chi-square test” estimation is better than the other two methods. A possible solution to the “zero recapture” problem is put forward. We find that if population size is at least a few thousand, two-sample CR estimation provides reasonable estimates of the population size.

Regression analysis is one of the most applied statistical techniques. The sta- tistical inference of a linear regression model with a monotone constraint had been discussed in early analysis. A natural question arises when it comes to the difference between the cases of with and without the constraint. Although the comparison be- tween confidence intervals of linear regression models with and without restriction for one predictor variable had been considered, this discussion for multiple regres- sion is required. In this thesis, I discuss the comparison of the confidence intervals between a multiple linear regression model with and without constraints.

Variable selection procedures for high dimensional data have been proposed and studied by a large amount of literature in the last few years. Most of the previous research focuses on the selection properties as well as the point estimation properties. In this paper, our goal is to construct the confidence intervals for some low-dimensional parameters in the high-dimensional setting. The models we study are the partially penalized linear and accelerated failure time models in the high-dimensional setting. In our model setup, all variables are split into two groups. The first group consists of a relatively small number of variables that are more interesting. The second group consists of a large amount of variables that can be potentially correlated with the response variable. We propose an approach that selects the variables from the second group and produces confidence intervals for the parameters in the first group. We show the sign consistency of the selection procedure and give a bound on the estimation error. Based on this result, we provide the sufficient conditions for the asymptotic normality of the low-dimensional parameters. The high-dimensional selection consistency and the low-dimensional asymptotic normality are developed for both linear and AFT models with high-dimensional data.

With new tests being developed and marketed, the comparison of the diagnostic accuracy of two continuous-scale diagnostic tests are of great importance. Comparing the partial areas under the receiver operating characteristic curves (pAUC) is an effective method to evaluate the accuracy of two diagnostic tests. In this thesis, we study the semi-parametric inference for the difference between two pAUCs. A normal approximation for the distribution of the difference between two pAUCs has been derived. The empirical likelihood ratio for the difference between two pAUCs is defined and its asymptotic distribution is shown to be a scaled chi-quare distribution. Bootstrap and empirical likelihood based inferential methods for the difference are proposed. We construct five confidence intervals for the difference between two pAUCs. Simulation studies are conducted to compare the finite sample performance of these intervals. We also use a real example as an application of our recommended intervals.

Compare the accuracy of two continuous-scale tests is increasing important when a new test is developed. The traditional approach that compares the entire areas under two Receiver Operating Characteristic (ROC) curves is not sensitive when two ROC curves cross each other. A better approach to compare the accuracy of two diagnostic tests is to compare the areas under two ROC curves (AUCs) in the interested specificity interval. In this thesis, we have proposed bootstrap and empirical likelihood (EL) approach for inference of the difference between two partial AUCs. The empirical likelihood ratio for the difference between two partial AUCs is defined and its limiting distribution is shown to be a scaled chi-square distribution. The EL based confidence intervals for the difference between two partial AUCs are obtained. Additionally we have conducted simulation studies to compare four proposed EL and bootstrap based intervals.

Credit Line Analysis plays a very important role in the housing market, especially with the situation of large number of frozen loans during the current financial crisis. In this thesis, we apply the methods of kernel estimate and the Receiver Operating Characteristic (ROC) curve in the credit loan application process in order to help banks select the optimal threshold to differentiate good customers from bad customers. Better choice of the threshold is essential for banks to prevent loss and maximize profit from loans. One of the main advantages of our study is that the method does not require us to specify the distribution of the latent risk score. We apply bootstrap method to construct the confidence interval for the estimate.

We take a Student process that is based on independent copies of a random variable X and has trajectories in the function space D[0,1]. As a consequence of a functional central limit theorem for this process, with X in the domain of attraction of the normal law, we consider convergence in distribution of several functionals of this process and derive respective asymptotic confidence intervals for the mean of X. We explore the expected lengths and finite-sample coverage probabilities of these confidence intervals and the one obtained from the asymptotic normality of the Student t-statistic, thus concluding some alternatives to the latter confidence interval that are shorter and/or have at least as high coverage probabilities.

Several methods have been proposed to estimate the kurtosis of a distribution. The three common estimators are: g2, G2 and b2. This thesis addressed the performance of these estimators by comparing them under the same simulation environments and conditions. The performance of these estimators are compared through confidence intervals by determining the average width and probabilities of capturing the kurtosis parameter of a distribution. We considered and compared classical and non-parametric methods in constructing these intervals. Classical method assumes normality to construct the confidence intervals while the non-parametric methods rely on bootstrap techniques. The bootstrap techniques used are: Bias-Corrected Standard Bootstrap, Efron’s Percentile Bootstrap, Hall’s Percentile Bootstrap and Bias-Corrected Percentile Bootstrap. We have found significant differences in the performance of classical and bootstrap estimators. We observed that the parametric method works well in terms of coverage probability when data come from a normal distribution, while the bootstrap intervals struggled in constantly reaching a 95% confidence level. When sample data are from a distribution with negative kurtosis, both parametric and bootstrap confidence intervals performed well, although we noticed that bootstrap methods tend to have smaller intervals. When it comes to positive kurtosis, bootstrap methods perform slightly better than classical methods in coverage probability. Among the three kurtosis estimators, G2 performed better. Among bootstrap techniques, Efron’s Percentile intervals had the best coverage.

Increased attention to statistical concepts has been a prevalent trend in revised mathematics curricula across grade levels. However, the preparation of secondary school mathematics educators has not received similar attention, and learning opportunities provided to these educators is oftentimes insufficient for teaching statistics well. The purpose of this study is to analyze pre-service teachers' conceptions about confidence intervals. This research inquired about statistical reasoning from the perspective of students majoring in mathematics education enrolled in an undergraduate statistics education course who have previously completed an introductory course in statistics. We found common misconceptions among pre-service teachers participating in this study. An unanticipated finding is that all the pre-service teachers believed that the construction of a confidence interval relies on a sampling distribution that does not contain every possible sample. Instead, they believed it is necessary to take multiple samples and build a distribution of their means. I called this distribution the Multi-Sample Distribution (MSD).