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In the past two decades, there have been calls for statistical reform in psychology. Three key concepts within reform are effect sizes, confidence intervals and statistical power. The aim of this thesis was to examine the use and knowledge of these particular concepts, to examine whether researchers are suitably equipped to incorporate them into their research. This thesis consists of five studies. Study 1 reviewed author guidelines across 100 psychology journals, to look for any statistical recommendations. Study 2 (n = 247) and Study 3 (n = 56) examined the use and knowledge of effect sizes using a questionnaire and online experiment. Study 4 surveyed psychology researchers on their use and knowledge of confidence intervals (n = 206). Similarly, Study 5 surveyed psychology researchers on their use and knowledge of power analyses and statistical power (n = 214). Typically, psychology journals expect authors to report effect sizes in their work, although there are fewer expectations related to confidence intervals. Power analyses are also frequently encouraged for sample size justification. Self-reported use of effect sizes, confidence intervals and power analyses was high, while common barriers to use included a lack of knowledge, a lack of motivation, and the influence of academic peers. While knowledge of effect sizes was quite high, they appear to only be understood in relatively limited contexts. In contrast, both confidence intervals and statistical power appear to be frequently misunderstood, and many researchers find power analysis calculations difficult. Researchers would benefit from increased education and support to encourage them to confidently adopt an assortment of statistics in their work, and more effort must be made to prevent statistical changes from becoming a new series of tick-box exercises that do not improve the integrity of psychological research.

Regression models describingthe dependence between a univariate response and a set of covariates play a fundamental role in statistics. In the last two decades, a tremendous effort has been made in developing flexible regression techniques such as generalized additive models(GAMs) with the aim of modelling the expected value of a response variable as a sum of smooth unspecified functions of predictors. Many nonparametric regression methodologies exist includinglocal-weighted regressionand smoothing splines. Here the focus is on penalized regression spline methods which can be viewed as a generalization of smoothing splines with a more flexible choice of bases and penalties. This thesis addresses three issues. First, the problem of model misspecification is treated by extending the instrumental variable approach to the GAM context. Second, we study the theoretical and empirical properties of the confidence intervals for the smooth component functions of a GAM. Third, we consider the problem of variable selection within this flexible class of models. All results are supported by theoretical arguments and extensive simulation experiments which shed light on the practical performance of the methods discussed in this thesis.

We study the Feasible Generalized Least-Squares (FGLS) estimation of the parameters of a linear regression model in which the errors are allowed to exhibit heteroskedasticity of unknown form and to be serially correlated. The main contribution is two fold; first we aim to demystify the reasons often advanced to use OLS instead of FGLS by showing that the latter estimate is robust, and more efficient and precise. Second, we devise consistent FGLS procedures, robust to misspecification, which achieves a lower mean squared error (MSE), often close to that of the correctly specified infeasible GLS. In the first chapter we restrict our attention to the case with independent heteroskedastic errors. We suggest a Lasso based procedure to estimate the skedastic function of the residuals. This estimate is then used to construct a FGLS estimator. Using extensive Monte Carlo simulations, we show that this Lasso-based FGLS procedure has better finite sample properties than OLS and other linear regression-based FGLS estimates. Moreover, the FGLS-Lasso estimate is robust to misspecification of both the functional form and the variables characterizing the skedastic function. The second chapter generalizes our investigation to the case with serially correlated errors. There are three main contributions; first we show that GLS is consistent requiring only pre-determined regressors, whereas OLS requires exogenous regressors to be consistent. The second contribution is to show that GLS is much more robust that OLS; even a misspecified GLS correction can achieve a lower MSE than OLS. The third contribution is to devise a FGLS procedure valid whether or not the regressors are exogenous, which achieves a MSE close to that of the correctly specified infeasible GLS. Extensive Monte Carlo experiments are conducted to assess the performance of our FGLS procedure against OLS in finite samples. FGLS achieves important reductions in MSE and variance relative to OLS. In the third chapter we consider an empirical application; we re-examine the Uncovered Interest Parity (UIP) hypothesis, which states that the expected rate of return to speculation in the forward foreign exchange market is zero. We extend the FGLS procedure to a setting in which lagged dependent variables are included as regressors. We thus provide a consistent and efficient framework to estimate the parameters of a general k-step-ahead linear forecasting equation. Finally, we apply our FGLS procedures to the analysis of the two main specifications to test the UIP.

As computing capabilities and cloud-enhanced data sharing has accelerated exponentially in the 21st century, our access to Big Data has revolutionized the way we see data around the world, from healthcare to investments to manufacturing to retail and supply-chain. In many areas of research, however, the cost of obtaining each data point makes more than just a few observations impossible. While machine learning and artificial intelligence (AI) are improving our ability to make predictions from datasets, we need better statistical methods to improve our ability to understand and translate models into meaningful and actionable insights. A central goal in the world of statistics and data science is the construction of linear regression models for continuous variables of interest. Often, our objective is to examine the impact of one or more explanatory variables, after adjusting for demographic variables or some other known/relevant covariate(s). While the traditional methodology uses a combination of partial F-tests and individual t-tests to determine statistical significance, we know that p-values obtained from such methods are heavily dependent on sample size. This is particularly problematic for large datasets or "overpowered" studies, where even the tiniest of effects will appear to be highly significant, or for extremely small datasets, where real effects may not reach statistical significance. The coefficient of partial determination (also known as partial R2) is widely used in the applied sciences to supplement hypothesis testing, but little work has been done to understand its statistical properties. In this dissertation, the exact, complete distribution of partial R2 is derived, accompanied by simulation studies and real-world data examples to show the advantages of adding coefficients of determination to the analysis of quantitative data models, regardless of sample size. Additionally, two novel inference methods are proposed for both R2 and partial R2, which build on these distributional results to provide better coverage and more focused intervals for models built using small- and medium-sized datasets.

In SEM, fit indexes provide useful information about how well a hypothesized model fits the population. Bootstrap has been applied to construct confidence internals for fit indexes. We proposed the most recent method for constructing confidence intervals (CIs) of fit indexes (Cheng & Wu, 2017): the bootstrap-test-based method. This dissertation includes the most popular bootstrap CI methods and the bootstrap-test-based method. In addition to the percentile bootstrap CI method used in Zhang and Savalei (2016), we also included other popular bootstrap CI methods. For all bootstrap CI methods, we explored their performances with and without the transformation proposed by Yuan, Hayashi, and Yanagihara (2007). In this process, we also solved computation difficulty for Studentized CI. The bootstrap-test-based method is improved by using alternative search statistics. As the previous approaches were not extended to nonnormal conditions, the CI estimation for fit indexes with nonnormal data are investigated for both bootstrap CI and bootstrap-test-based methods, using adjusted definitions of fit indexes for nonnormal data. Different nonnormal data generation techniques are applied. This dissertation presents a comprehensive comparison of bootstrap CI methods and the bootstrap-test-based method under various conditions. From the simulation results, the CIs for fit indexes under the bootstrap-test-based method are more accurate than most bootstrap CI methods. The results also show that the bootstrap-test-based method can be generalized well to non-normal data. The confidence bounds coverage by bootstrap-test-based method are closer to their nominal levels.

In a cluster randomized trial (CRT), groups of people are randomly assigned to receive different interventions. CRTs are useful when interventions are more naturally or feasibly applied at the cluster level or when we wish to quantify population-level effects. Existing parametric and semiparametric statistical methods for analyzing CRTs can perform poorly when distributional assumptions are violated, when there are a small number of clusters, or when some outcomes are missing. This dissertation is devoted to developing robust statistical methods to address these challenges. First, we focus on randomization-based inference, an alternative approach for analyzing CRTs that is distribution-free and does not require a large number of clusters to be valid. Although it is well known that a confidence interval (CI) can be obtained by inverting a randomization test, this requires testing a non-zero null hypothesis, which is challenging with non-continuous and survival outcomes. In Chapter 1, we introduce a general method for calculating randomization-based CIs in parallel CRTs using regression models with fixed offset terms. This approach accommodates various outcome types and employs an efficient algorithm to speed up CI computation. In Chapter 2, we extend this methodology to a stepped wedge trial (SWT), a CRT design in which clusters crossover from control to intervention at different time points. Here, we also examine how our method compares to existing randomization-based approaches for SWTs based on cluster-period summary statistics, and demonstrate how to account for design features---and why it is important to do so---in a randomization-based analysis. In Chapter 3, we investigate multiply robust (MR) estimation in CRTs with missing outcomes. Unlike imputation, inverse probability weighting, or doubly robust estimation, MR estimation allows for various sets of multiple models (e.g. for the propensity scores, covariate-adjusted means) to be specified, with consistency guaranteed if any one of these models is correct. We establish MR estimators for the marginal mean, scale, and correlation parameters (e.g. the intracluster correlation coefficient) in a CRT via weighted generalized estimating equations.

Many studies deal with inflated and nonnegative data, such as in medical studies. Most studies that deal with inflated data deal with zero-inflated datasets, but there are many datasets that are zero-one inflated as well. Zero-inflated datasets are characterized by a significant proportion of zero values, leading to a skewed distribution. Zero-One inflated datasets are characterized by a significant proportion of zero and one values, which also leads to a skewed distribution. It is common practice to use the Central Limit Theorem to assume an approximately normal distribution to construct confidence intervals and conduct hypothesis tests. However with inflated and highly skewed distributions, this practice leads to an inaccurate result. The empirical likelihood method offers an alternative method of computing confidence intervals with the benefit of having no distributional assumptions. Although the empirical likelihood method provides an improvement, it suffers from several drawbacks.In this dissertation, we propose several modified empirical likelihood methods to combat these drawbacks. We use these modified methods, along with the empirical likelihood and normal approximation methods, to construct confidence intervals based on zero-inflated data and zero-one inflated data. We compare the performance of each method for these two situations on both simulated data and real data. Furthermore, we develop a hypothesis test for comparing two means based on one of the modified empirical likelihood approaches. We then test the modified empirical likelihood approach against the empirical likelihood and normal approximation methods using simulated and real data.

Distance correlation is a measure of the relationship between random vectors in arbitrary dimension. A sample distance covariance can be formulated in both an unbiased estimator and a biased estimator of distance covariance, where distance correlation is defined as the normalized coefficient of distance covariance. The jackknife empirical likelihood for a U-statistic by Jing, Yuan, and Zhou (2009) can be applied to a distance correlation since the empirical likelihood method fails in nonlinear statistics. A Wilks' theorem for jackknife empirical likelihood is shown to hold for distance correlation. This research shows how to construct a confidence interval for distance correlation based on jackknife empirical likelihood for a U-statistic, where the sample distance covariance can be represented as a U-statistic. In comparing coverage probabilities of confidence intervals for distance correlation based on jackknife empirical likelihood and bootstrap method, coverage probabilities for the jackknife empirical likelihood show more accuracy. We propose the estimation and the visualization of local distance correlation by using a local version of the jackknife empirical likelihood. The kernel density functional estimation is used to construct the jackknife empirical likelihood locally. The bandwidth selection for kernel function should minimize the distance between the true density and estimated density. Local distance correlation has the property that it equals zero in the neighborhood of each point if and only if the two variables are independent in that neighborhood. The estimation and visualization of local distance correlation are shown as accurate to capture the local dependence when compared with the local Gaussian correlation in simulation studies and real examples.

In the big data era, regression models with a large number of covariates have emerged as a common tool to tackle problems arising from business, engineering, genomics, neuroimaging, and epidemiological studies. Drawing statistical inference for these models has sparked much interest over the past few years. Albeit successful for high dimensional linear models, high dimensional inference approaches beyond linear regression are limited and present unsatisfactory performance, theoretically or numerically. In this dissertation, we focus on de-biased lasso, which has been one of the most popular methods for high dimensional inferences. We propose procedures that provide better bias correction and confidence interval coverage, and draw reliable inference for regression parameters in the "large n, diverging p" scenario. In general, we caution against applying de-biased lasso and its variants to models beyond linear regression when parameters outnumber the sample size. Following an overview outlined in Chapter I, we focus on the generalized linear models (GLMs) in Chapter II. Extensive numerical simulations indicate that de-biased lasso may not adequately remove biases for high dimensional GLMs, and thus yield unreliable confidence intervals. We have further found that several key assumptions, especially the sparsity condition on the inverse Hessian matrix, may not hold for GLMs. In a "large n, diverging p" scenario, we consider an alternative de-biased lasso approach that inverts the Hessian matrix of the concerned model without requiring matrix sparsity, and establish the asymptotic distributions of linear combinations of the estimates. Simulations evidence that our proposed de-biased estimator performs better in bias correction and confidence interval coverage for a wide range of p/n ratios. We apply our method to the Boston Lung Cancer Study, an epidemiology study on the mechanisms underlying lung cancer, and investigate the joint effects of genetic variants on overall lung cancer risks. In Chapter III, we draw inference based on the Cox proportional hazards model with a diverging number of covariates. As the existing methods assume sparsity on the inverse of the Fisher information matrix, which may not hold for Cox models, they typically generate biased estimates and under-covered confidence intervals. We modify de-biased lasso by using quadratic programming to approximate the inverse of the information matrix, without posing matrix sparsity assumptions. We establish the asymptotic theory for the estimated regression coefficients when the covariate dimension diverges with the sample size. With extensive simulations, our proposed method provides consistent estimates and confidence intervals with improved coverage probabilities. We apply the proposed method to assess the effects of genetic markers on overall survival of non-small cell lung cancer patients in the aforementioned Boston Lung Cancer Study. Stratified Cox proportional hazards model, with extensive applications in large scale cohort studies, are useful when some covariates violate the proportional hazards assumption or data are stratified based on factors, such as transplant centers. In Chapter IV, we extend the de-biased lasso approach proposed in Chapter III to draw inference for the stratified Cox model with potentially many covariates. We provide asymptotic results useful for inference on linear combinations of the regression parameters, and demonstrate its utility via simulation studies. We apply the method to analyze the national kidney transplantation data stratified by transplant center, and assess the effects of many factors on graft survival.

We study interval estimation for parameters of discrete distributions, focusing on thebinomial, Poisson, negative binomial, and hypergeometric distributions explicitly. Weprovide a broad treatment of the problem, covering both conventional and randomizedconfidence intervals, as well as Geyer and Meeden’s concept of fuzzy confidence intervals.We take a graphical approach to the problem through the use of coverage probabilityfunctions and determine the optimal procedure under each of a wide variety of criteria,including multiple notions of length. Several new methods are proposed, including amethod that produces length optimal fuzzy confidence intervals. Credible intervals andmulti-parameter discrete distributions are also considered.

Generalizations of univariate distributions are often of interest to serve for real life phenomena. These generalized distributions are very useful in many ¿elds such as medicine, physics, engineer-ing and biology. Lomax distribution (Pareto-II) is one of the well known univariate distributions that is considered as an alternative to the exponential, gamma, and Weibull distributions for heavy tailed data. However, this distribution does not grant great ¿exibility in modeling data.In this dissertation, we introduce a generalization of the Lomax distribution called Rayleigh Lo-max (RL) distribution using the form obtained by El-Bassiouny et al. (2015). This distribution provides great ¿t in modeling wide range of real data sets. It is a very ¿exible distribution that is related to some of the useful univariate distributions such as exponential, Weibull and Rayleigh dis-tributions. Moreover, this new distribution can also be transformed to a lifetime distribution which is applicable in many situations. For example, we obtain the inverse estimation and con¿dence intervals in the case of progressively Type-II right censored situation. We also apply Schwartz information approach (SIC) and modi¿ed information approach (MIC) to detect the changes in parameters of the RL distribution. The performance of these approaches is studied through simu-lations and applications to real data sets.According to Aryal and Tsokos (2009), most of the real world phenomenon that we need to study are asymmetrical, and the normal model is not a good model for studying this type of dataset. Thus, skewed models are necessary for modeling and ¿tting asymmetrical datasets. Azzalini (1985) in-troduced the univariate skew normal distribution and his approach can be applied in any symmet-rical model. However, if the underlying (base) probability is not symmetric, we can not apply the Azzalini’s approach. This motivated the study for more ¿exible alternative.Shaw and Buckley (2007) introduced a quadratic rank transmutation map (QRTM) which can be applied in any (symmetric or asymmetric) distribution. Recently, many distributions have been suggested using the QRTM to derive the transmuted class (TC) of distributions. This provides great ¿exibility in performing real datasets. We extend our work in RL distribution to derive the transmuted Rayleigh Lomax (TR-RL) distribution using the QRTM. Mathematical and statistical properties, such as moment generating function, L-moment, probability weight moments are de-rived and studied. We also establish the relationship between the TR-RL , the RL, and other useful distributions to show that our proposed distribution includes them as special cases. TR-RL is ¿tted to a well known dataset, the goodness of ¿t test and the likelihood ratio test are presented to show how well the TR-RL ¿ts the data.

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.

The aim of this research is to investigate and develop methods for building confidence intervals (CIs) for parameter functions of discrete choice models, with a special focus on the CIs for willingness-to-pay measures. CIs are more than simply statistical measures. Rather, they are a convenient and easily understood means by which the variability of a parameter or sample statistic can be reported, especially because they can be presented graphically. CIs should be reported for all random statistics, and especially so in applied work where one cannot assume that the estimated parameter would exactly equal the true (unknown) parameter. Yet, when presenting willingness-to-pay values, the CIs are often neglected. This is partially because building CIs for willingness-topay values is not a trivial task, due to the possibility of discontinuity in the willingness-to-pay measure and its unknown probability distribution a priori. In addition, the methods used to build these intervals are debated greatly, with no consensus as to the best method to use. This research consolidates the contradictory results and presents reasons for the disparity currently present in the literature. It also extends the work of building CIs beyond willingness-to-pay measures to other parameter functions; in particular, this research demonstrates how CIs can be built for the probability that an airline passenger cancels his ticket. The methods of building CIs are studied using Monte Carlo simulations and case studies. Results indicate that when sample sizes or the price parameter is large (i.e. there are fewer chances for discontinuity to occur), all the preference space methods studied work equally well. However, under weak identification (when the price parameter is small), the Fieller method performs best. Hence, in general, the Fieller method should be the preferred method for building CIs for willingness-to-pay values. This research also proposes the use of the Bayesian post-processing method to build CIs. This method, though a viable option, is not often discussed. The Bayesian method also has an edge over the other methods studied for several reasons, including the ease of constructing individual CIs and the ability to incorporate factors such as historical data into the model.

Utilities operating nuclear power plants in the United States are required to demonstrate that their plants comply with the safety requirements set by the U.S. Nuclear Regulatory Commission (NRC). How to show adherence to these limits through the use of computer code surrogates is not always straightforward, and different techniques have been proposed and approved by the regulator. The issue of compliance with regulatory limits is examined by rephrasing the problem in terms of hypothesis testing. By using this more rigorous framework, guidance is proposed to choose techniques to increase the probability of arriving at the correct conclusion of the analysis. The findings of this study show that the most straightforward way to achieve this goal is to reduce the variance of the output result of the computer code experiments. By analyzing different variance reduction techniques, and different methods of satisfying the NRC’s requirements, recommendations can be made about the best-practices, that would result in a more accurate and precise result. This study began with an investigation into the point estimate of the 0.95-quantile using traditional sampling methods, and new orthogonal designs. From there, new work on how to establish confidence intervals for the outputs of experiments designed using variance reduction techniques was compared to current, regulator-approved methods. Lastly, a more direct interpretation of the regulator’s probability requirement was used, and confidence intervals were established for the probability of exceeding a safety limit. From there, efforts were made at combining methods, in order to take advantage of positive aspects of different techniques. The results of this analysis show that these variance reduction techniques can provide a more accurate and precise result compared to current methods. This means an increased probability of arriving at the correct conclusion, and a more accurate characterization of the risk associated with events. While several of these methods are asymptotic in nature, which presents potential drawbacks, issues of convergence appear to be outweighed by the reduction in variance, and improvement of the information contained in the results. Using this knowledge, recommendations were made about the applicability of these methods in the field of reactor safety, and about future regulatory limits and their implications.

In this paper we consider the problem of constructing confidence intervals and lower bounds forthe intraclass correlation coefficient in an interrater reliability study where the raters are randomly selected from a population of raters.The likelihood function of the interrater reliability is derived and simplified, and the profile likelihood based approach is readily available for computing the confidence intervals of the interrater reliability. Unfortunately, the confidence intervals computed by using the profile likelihood function are in general too narrow to have the desired coverage probabilities. From the point view of practice, a conservative approach, if is at least as precise as any existing method, is preferred sinceit gives the correct results with a probability higher than claimed. Under this rationale, we propose the so-called modified likelihood approach in this paper. Simulation study shows that, the proposed method in general has better performance than currently used methods.

The objective of this thesis is to examine one of the most fundamental and yet important methodologies used in statistical practice, interval estimation of the probability of success in a binomial distribution. The textbook confidence interval for this problem is known as the Wald interval as it comes from the Wald large sample test for the binomial case. It is generally acknowledged that the actual coverage probability of the standard interval is poor for values of p near 0 or 1. Moreover, recently it has been documented that the coverage properties of the standard interval can be inconsistent even if p is not near the boundaries. For this reason, one would like to study the variety of methods for construction of confidence intervals for unknown probability p in the binomial case. The present thesis accomplishes the task by presenting several methods for constructing confidence intervals for unknown binomial probability p. It is well known that the hypergeometric distribution is related to the binomial distribution. In particular, if the size of the population, N, is large and the number of items of interest k is such that k/N tends to p as N grows, then the hypergeometric distribution can be approximated by the binomial distribution. Therefore, in this case, one can use the confidence intervals constructed for p in the case of the binomial distribution as a basis for construction of the confidence intervals for the unknown value k = pN. The goal of this thesis is to study this approximation and to point out several confidence intervals which are designed specifically for the hypergeometric distribution. In particular, this thesis considers several confidence intervals which are based on estimation of a binomial proportion as well as Bayesian credible sets based on various priors.

Accurate clock synchronization is essential in the operation of networks. In applications in which it can be performed frequently, the process generally relies upon the estimation of the relative offset of two different clocks. However, due to limited power resources in Wireless Sensor Networks, it cannot be performed often. Thus in order to obtain a longer lasting synchronization, the relative frequency of two clocks is also estimated. In the literature, there are estimators for relative offset and frequency based on a variety of techniques including graphical methods, maximum likelihood estimation, and least squares estimation. However, there has been no extensive study to compare the performance of these estimators. In this dissertation, the existing estimators are examined in detail and several alternative least squares-based estimators are developed and compared with the existing estimators in an exhaustive study.Although there is a large literature on estimating relative offset and frequency, there is little to no research regarding inference. This topic is of potentially great importance because it may possibly lead to limiting unneeded synchronization, resulting in better conservation of network power resources. Recent work has focused on the construction of confidence intervals for relative offset and while it does not address relative frequency, it is an important first step towards developing procedures that do. In this work, it has been assumed that the network delays that occur during the synchronization process are independent, but this may not be appropriate in all applications. The network delays are often modeled as independent exponential random variables. Thus to improve upon the existing confidence interval methodology, in this dissertation the use of bivariate exponential distributions is introduced to capture the anticipated correlation between the network delays. Moreover, an alternative confidence interval procedure for offset is developed and is then used to illustrate how the assumption of independent network delays can potentially lead to improper inference about relative offset.

To analyze real-world events, researchers collect observation data from an underlying process and construct models to represent the observed situation. In this work, we consider issues that affect the construction and usage of a specific type of model. Markov models are commonly used because their combination of discrete states and stochastic transitions is suited to applications with both deterministic and stochastic components. Hidden Markov Models (HMMs) are a class of Markov model commonly used in pattern recognition. We first demonstrate how to construct HMMs using only the observation data, and no a priori information, by extending a previously developed approach from J.P. Crutchfield and C.R. Shalizi. We also show how to determine with a level of statistical confidence whether or not the model fully encapsulates the underlying process. Once models are constructed from observation data, the models are used to identify other types of observations. Traditional approaches consider the maximum likelihood that the model matches the observation, solving a classification problem. We present a new method using confidence intervals and receiver operating characteristic curves. Our method solves a detection problem by determining if observation data matches zero, one, or more than one model. To detect the occurrence of a behavior in observation data, one must consider the amount of data required. We consider behaviors to be 'serial Markovian,' when the behavior can change from one model to another at any time. When analyzing observation data, considering too much data induces high delay and could lead to confusion in the system if multiple behaviors are observed in the data stream. If too little data is used, the system has a high false positive rate and is unable to correctly detect behaviors. We demonstrate the effectiveness of all methods using illustrative examples and consumer behavior data.

Youden index, a main summary index for the Receiver Operating Characteristic (ROC) curve, is a comprehensive measurement for the effectiveness of a diagnostic test. For a continuous-scale diagnostic test, the optimal cut-point for the positive of disease is the cut-point leading to the maximization of the sum of sensitivity and specificity. Finding the Youden index of the test is equivalent to maximize the sum of sensitivity and specificity for all the possible values of the cut-point. In this thesis, we propose a new non-parametric confidence interval for the Youden index. Extensive simulation studies are conducted to compare the relative performance of the new interval with the existing intervals for the index. Our simulation results indicate that the newly developed non-parametric method performs as well as the existing parametric method but it has better finite sample performance than the existing non-parametric methods. The new method is flexible and easy to implement in practice. A real example is also used to illustrate the application of the proposed interval.

Lorenz curves are extensively used in economics to analyze income inequality metrics. In this thesis, we discuss confidence interval estimation methods for generalized Lorenz curve. We first obtain normal approximation (NA) and empirical likelihood (EL) based confidence intervals for generalized Lorenz curves. Then we perform simulation studies to compare coverage probabilities and lengths of the proposed EL-based confidence interval with the NA-based confidence interval for generalized Lorenz curve. Simulation results show that the EL-based confidence intervals have better coverage probabilities and shorter lengths than the NA-based intervals at 100p-th percentiles when p is greater than 0.50. Finally, two real examples on income are used to evaluate the applicability of these methods: the first example is the 2001 income data from the Panel Study of Income Dynamics (PSID) and the second example makes use of households’ median income for the USA by counties for the years 1999 and 2006

We consider optimization problems in the presence of uncertainty for three different scientific examples. The first optimization problem addressed is in the area of parameter estimation. We review the asymptotic theory for standard errors in classical ordinary least squares (OLS) inverse or parameter estimation problems involving general nonlinear dynamical systems where sensitivity matrices can be used to compute the asymptotic covariance matrices. We discuss possible pitfalls in computing standard errors in regions of low parameter sensitivity and/or near a steady state solution of the underlying dynamical system. Next we consider electromagnetic evasion-interrogation games where the evader can use ferroelectric material coatings to avoid detection while the interrogator can manipulate the interrogating frequencies and angles of incidence to enhance detection. Each player in this two-player game wishes to change the amount of reflected signal created in a way that will benefit them the most. Thus both players will attempt to optimize their chances of either remaining undetected or detecting their opponent. With the introduction of uncertainty, the resulting game is carried out over spaces of probability measures. Finally, a one player dynamical game is formulated. The premise here is for the evader to manipulate the ferroelectric material coatings to avoid detection based upon updated information about the frequencies being sent out. The uncertainty is found in the frequencies that the interrogator will employ. We incorporate both drift and diffusion into this optimization problem.

Diagnostic testing is essential to distinguish non-diseased individuals from diseased individuals. More accurate tests lead to improved treatment and thus reduce medical mistakes. The sensitivity and specificity are two important measurements for the diagnostic accuracy of a diagnostic test. When the test results are continuous, it is of interest to construct a confidence interval for the sensitivity at a fixed level of specificity for the test. In this thesis, we propose three empirical likelihood intervals for the sensitivity. Simulation studies are conducted to compare the empirical likelihood based confidence intervals with the existing normal approximation based confidence interval. Our studies show that the new intervals had better coverage probability than the normal approximation based interval in most simulation settings.

The mean residual life (MRL) function is an important function in survival analysis, actuarial science, economics and other social sciences and reliability for characterizing lifetime. Different methods have been proposed for doing inference on the MRL but their coverage probabilities for small sample sizes are not good enough. In this thesis we apply the empirical likelihood method and carry out a simulation study of the MRL function using different statistical distributions. The simulation study does a comparison of the empirical likelihood method and the normal approximation method. The comparisons are based on the average lengths of confidence intervals and coverage probabilities. We also did comparisons based on median lengths of confidence intervals for the MRL. We found that the empirical likelihood method gives better coverage probability and shorter confidence intervals than the normal approximation method for almost all the distributions that we considered. Applying the two methods to real data we also found that the empirical likelihood method gives thinner pointwise confidence bands.

As more and more dense, yet cost-effective, genetic maps become increasingly available, the focus of linkage analysis is shifting from testing for linkage signals to sufficiently localizing putative disease loci before fine mapping begins. Currently, there exists only a limited number of methods that provide confidence regions for the locations of trait loci. Among them is the confidence set inference (CSI) procedure based on the mean IBD sharing statistic for data from affected sib-pair studies described by Lin (2002) that deduces such regions with known lower bound on their coverage. Although this method has many attractive features, including avoidance of multiplicity adjustment for the number of markers scanned, its formulation poses some restrictions that limit its usefulness on practical applications. First, it assumes that all markers are 100% polymorphic, so that the IBD state at each of them is inferred unequivocally, an assumption rarely met in reality. Second, when the genetic map available is sparse, it tends to produce intervals that overcover the trait locus. Finally, its application requires knowledge of the IBD sharing distribution at the trait locus by an affected sib-pair. These probabilities are estimated using population disease characteristics that can be obtained through epidemiological studies with reasonable accuracy. However, there is a number of issues that renders this method of estimating the IBD distribution impractical. We propose several extensions that address some of the limitations of the CSI approach. First, we extend it to accommodate markers with incomplete polymorphism, thereby increasing its practical value. Next, we modify it so that it tests each location on the genome for its possibility to be the trait locus. This way, we obtain regions with known exact coverage probability, rather than placing a lower bound on it. Finally, a two-step application of the CSI approach promises to avoid using population disease characteristics, circumventing the issues associated with them. Through extensive simulations, theoretical results, and applications to real data we demonstrate that the new CSI versions are indeed effective tools for localizing genes, with increased capability, when compared to currently employed approaches.

Several statistical methods, used to construct confidence intervals (CIs) for a single Binomial proportion, are selected based on literature recommendations and are compared. A new criterion ‘strict nesting condition’ is defined when comparing different confidence interval methods. The focus is on Blaker’s method and its potential shortness in practice is discussed. The continuity correction and smoothing technique are proposed to improve Blaker’s method. The fundamental confidence interval obtained from a fundamentally defined p-value is also introduced and compared with the Blaker’s and Clopper-Pearson’s (C-P’s) methods. The continuity correction (CC) and smoothing technique can also be used to improve the fundamental method. The modified fundamental method and Blaker’s method have great similarities in term of coverage, length, confidence curve shape, etc. The power behaviors in modified CC Blaker’s and fundamental methods are examined and compared with C-P’s. For two sample proportion inference, the three existing exact confidence interval methods that are available in StatXact are discussed and compared for the difference of 2 Binomial proportions. A simple approach based on the one sample exact methods is introduced to obtain the exact confidence interval of the difference when data are balanced. Its performance is comparable to these existing methods but the computation is much simpler.

The time it will take to run a program on a large problem size is estimated by sampling several smaller instances of the problem. Confidence intervals for the prediction are included. A time budget for sampling is used, so the estimate can be made quickly (but less precisely) or more slowly with greater accuracy. How to pick the sample points optimally under some simple assumptions is derived and proven. Overall accuracy is comparable to simulation or instruction counting, with much faster results; however, the results only apply to the given machine, not other platforms.

The dissertation considers construction of confidence intervals for a cumulative distribution function F(z) and its inverse at some fixed points z and u on the basis of an i.i.d. sample where the sample size is relatively small. The sample is modeled as having the flexible Generalized Gamma distribution with all three parameters being unknown. This approach can be viewed as an alternative to nonparametric techniques which do not specify distribution of X and lead to less efficient procedures. The confidence intervals are constructed by objective Bayesian methods and use the Jeffreys noninformative prior. Performance of the resulting confidence intervals is studied via Monte Carlo simulations and compared to the performance of nonparametric confidence intervals based on binomial proportion. In addition, techniques for change point detection are analyzed and further evaluated via Monte Carlo simulations. The effect of a change point on the interval estimators is studied both analytically and via Monte Carlo simulations.

Despite strong theoretical reasons why indications of uncertainty should be valuable to decision-making, Uncertainty Information (UI) is arbitrarily utilised in practice. Moreover, it’s unclear whether UI improves decisions. This thesis seeks to understand under what conditions UI has utility by defining a contingent model of UI utility, and testing part of the model. This will help establish whether UI should be used more in practice, and if so, how. Theoretical and empirical literature was reviewed to develop a prescriptive model summarising the contingent utility of UI, which was used to design three paper-based laboratory experiments. Experiment 1 investigated UI utility with a capital investment task. For several investment opportunities participants determined whether achievement of a target level was at least 75% certain. Those participants provided with UI in the form of a 75% confidence level accompanied by a forecast made better decisions. Those with UI had performance compromised when separation of the target level and 75% confidence level was minimal. This may have been caused by: the forecast-confidence level range indicating the decision objective was being threatened; perception the UI contained some error; or, indifference to the situation. Using a similar task, Experiment 2 introduced a fourth treatment group, which received UI without a forecast, and included an additional factor by varying task uncertainty. Results agreed with Experiment 1. The decision performance of the group receiving UI without a forecast improved as task uncertainty increased, probably because the main decision rule used in that group was dependent upon the visual distance between the 75% confidence level and target level. Those with UI had better calibrated decision confidence. Experiment 3 was identical to Experiment 2, except presentation scale was varied to investigate against the effects of task uncertainty. Results agreed with Experiment 2, and presentation scale had no impact on UI utility. The main findings of this thesis demonstrate that UI has practical value given the right circumstances. These findings provide justification to those disciplines already utilising UI, and emphasise that those disciplines not utilising UI, such as management and accounting, do so against normative value and practical evidence.

Recent experiments in high energy physics have sparked interest in problems where a parameter is restricted to a portion of its natural range-for example, a positive normal mean. A recent article has shown that the Bayesian credible intervals that arise when the parameter is given a uniform distribution over the restricted range have good frequentist coverage probabilities. Here the latter result is examined for robustness when nuisance parameters are included in the model. It is shown that the frequentist coverage probabilities are still high, but there is a mild surprise in the nature of the intervals. Consider the model X = B + S, where the background B and the signal S are independent Poisson random variables with parameter b and theta, b is known, but theta is not. The model arises in high-energy physics, and some recent articles have suggested conditioning on the observed bound for B; that is, if X = n is observed, then the suggestion is to base the inference on the conditional distribution of X given B ≤ n. This suggestion is used here to derive a predictor of the signal, and the predictor is shown to be admissible and minimax. Classical confidence intervals for the ratio of variance components can degenerate to a single point with positive probability, leaving no allowance for error. An alternative method is considered which avoids this problem, but maintains the frequentist coverage probability. The alternative intervals are then compared to Bayesian credible intervals, derived from a non-informative prior; and the Bayesian intervals are shown to have quite high frequentist coverage probabilities.

The objective of this thesis is to estimate, for a distribution network, confidence intervals containing the values of nodal hourly power consumption and nodal maximum power consumption per customer where they are not measured. The values of nodal hourly power consumption are needed in operational as well as in planning stages to carry out load flow studies. As for the values of nodal maximum power consumption per customer, they are used to solve planning problems such as transformer sizing. Confidence interval estimation was preferred to point estimation because it takes into consideration the large variability of the consumption values. A computationally intensive statistical technique, namely the bootstrap method, is utilized to estimate these intervals. It allows us to replace idealized model assumptions for the load distributions by model free analyses. Two studies have been executed. The first one is based on the original nonparametric bootstrap method to calculate a 95% confidence interval for nodal hourly power consumption. This estimation is carried out for a given node and a given hour of the year. The second one makes use of the parametric bootstrap method in order to infer a 95% confidence interval for nodal maximum power consumption per customer. This estimation is realized for a given node and a given month. Simulation results carried out on a real data set are presented and discussed.

This research examined the performance of three parametric methods for confidence intervals: the classical, the Bonferroni, and the bootstrap-t method, as applied to estimating the mean of voucher populations in accounting. Usually auditing populations do not follow standard models. The population for accounting audits generally is a nonstandard mixture distribution in which the audit data set contains a large number of zero values and a comparatively small number of nonzero errors. This study assumed a situation in which only overstatement errors exist. The nonzero errors were assumed to be normally, exponentially, and uniformly distributed. Five indicators of performance were used. The classical method was found to be unreliable. The Bonferroni method was conservative for all population conditions. The bootstrap-t method was excellent in terms of reliability, but the lower limit of the confidence intervals produced by this method was unstable for all population conditions. The classical method provided the shortest average width of the confidence intervals among the three methods. This study provided initial evidence as to how the parametric bootstrap-t method performs when applied to the nonstandard distribution of audit populations of line items. Further research should provide a reliable confidence interval for a wider variety of accounting populations.