The purpose of this study was to correct some mistakes in the literature and derive a necessary and sufficient condition for the MRL to follow the roller-coaster pattern of the corresponding failure rate function. It was also desired to find the conditions under which the discrete failure rate function has an upside-down bathtub shape if corresponding MRL function has a bathtub shape. The study showed that if discrete MRL has a bathtub shape, then under some conditions the corresponding failure rate function has an upside-down bathtub shape. Also the study corrected some mistakes in proofs of Tang, Lu and Chew (1999) and established a necessary and sufficient condition for the MRL to follow the roller-coaster pattern of the corresponding failure rate function. Similarly, some mistakes in Gupta and Gupta (2000) are corrected, with the ensuing results being expanded and proved thoroughly to establish the relationship between the crossing points of the failure rate and associated MRL functions. The new results derived in this study will be useful to model various lifetime data that occur in environmental studies, medical research, electronics engineering, and in many other areas of science and technology.

Employing the approach proposed by Z. Chen for constructing an exact confidence interval for the location parameter, this study has investigated the exact confidence intervals, confidence limits and point estimators for the location parameter μ of the three-parameter Weibull distributions. Statistical simulation was carried out for different selections of i, j and k with specified confidence level and sample size. The critical values (ωα/2 and ω1-α/2 have been found using Mote-Carlo simulation. The optimization of the combination of i, j and k has been discussed. The point estimator for the location parameter of the three-parameter Weibull distributions is explored. It is observed that the critical values do not depend on the parameters. Simulation results show that the optimization of i, j and k is i=1, k-n and j=[n+2/3]. Compared with the commonly used MLE method, the described method provides a simpler, more accurate and more efficient way to estimate the location parameter of the three-parameter Weibull distributions. The described method yields very good statistical inferences for the location parameter of the three-parameter Weibull distributions.

In ecological risk assessment (ERA), it is important to know whether the exposure that animal species receive from a chemical concentration exceeds the desired safety level. This study examined several statistical methods currently being used in ecological risk assessment and reviewed several statistical procedures related to this subject in the literature. Two large sample nonparametric tests were developed for this study. Monte Carlo study showed that these tests performed well even when the sample size was moderately large. A real data set was used to show that the new methodologies provide a good method for assessing the potential risks of pesticides residues at an investigated site.

This thesis proposes some confidence intervals for the mean of a positively skewed distribution. The following confidence intervals are considered: Student-t, Johnson-t, median-t, mad-t, bootstrap-t, BCA, T1 , T3 and six new confidence intervals, the median bootstrap-t, mad bootstrap-t, median T1, mad T1 , median T3 and the mad T3. A simulation study has been conducted and average widths, coefficient of variation of widths, and coverage probabilities were recorded and compared across confidence intervals. To compare confidence intervals, the width and coverage probabilities were compared so that smaller widths indicated a better confidence interval when coverage probabilities were the same. Results showed that the median T1 and median T3 outperformed other confidence intervals in terms of coverage probability and the mad bootstrap-t, mad-t, and mad T3 outperformed others in terms of width. Some real life data are considered to illustrate the findings of the thesis.

In this thesis we are interested in the impact of economic and financial factors, such as interest rate, tax payment, reinsurance, and investment return, on insurance business. The underlying risk models of insurance business that we consider range from the classical compound Poisson risk model to the newly emerging and more general Lévy risk model. In these risk models, we assume that the claim-size distribution belongs to some distribution classes according to its asymptotic tail behavior. We consider both light-tailed and heavy-tailed cases. Our study is through asymptotic tail probabilities. Firstly, we study the asymptotic tail probability of discounted aggregate claims in the renewal risk model by introducing a constant force of interest. In this situation we focus on claims with subexponential tails. We derive for the tail probability of discounted aggregate claims an asymptotic formula, which holds uniformly for finite time intervals. For various special cases, we extend this uniformity to be valid for all time horizons. Then, we investigate the asymptotic tail probability of the maximum exceedance of a sequence of random variables over a renewal threshold. We derive a unified asymptotic formula for this tail probability for both light-tailed and heavy-tailed cases. By using the previous result, we study how to capture the impact of tax payments on the ruin probability in the Lévy risk model. We introduce periodic taxation under which the company pays tax at a fixed rate on its net income during each period. Assuming the Lévy measure, representing the claim-size distribution in the Lévy risk model, has a subexponential tail, a convolution-equivalent tail, or an exponential-like tail, we derive for the ruin probability several explicit asymptotic relations, in which the prefactor varies with the tax rate, reflecting the impact of tax payments. Finally, we consider the renewal risk model in which the surplus is invested into a portfolio consisting of both a riskless bond and a risky stock. The price process of the stock is modeled by an exponential Lévy process. We derive an asymptotic formula for the tail probability of the stochastically discounted net loss process.

The subject area of this thesis is the theory of random walks on groups. First, we study random walks on products of hyperbolic groups and show that the Poisson boundary can be identified with an appropriate geometric boundary (the skeleton). Second, we show that in the particular case of free and free-product factors, the Hausdorff dimension of the conditional measures on product fibers of the Poisson boundary is related to the asymptotic entropy and the rate of escape of the corresponding conditional random walks via a generalized entropy-dimension formula.

In this thesis we study the matter of hypersensitivity to model structure in the Rosenzweig- MacArthur predator-prey model, and in particular whether the introduction of stochasticity reduces the sensitivity of the !-limit sets to small changes in the underlying vector field. To do this, we study the steady-state probability distributions of stochastic differential equations driven by a compound Poisson process on a bounded subset of Rn, as steady-state distributions are analogous to !-limit sets for stochastic differential equations. We take a primarily analytic approach, showing that the steady-state distributions are equivalent to weak measure-valued solutions to a certain partial differential equation. We then analyze perturbations of the underlying vector field using tools from the theory of compact operators. Finally, we numerically simulate and compare solutions to both the deterministic and stochastic versions of the Rosenzweig-MacArthur model. / Thesis / Master of Science (MSc)

This dissertation has three principle areas of research: mixed type random variables, confidence regions, and golden quantile rank sets. While each offers a specific focus, some common themes persist; broadly stated, there are three. First, computational graphics play a critical role. Second, software development facilitates implementation and accessibility. Third, statistical analysis---often attributable to the aforementioned automation---provides valuable insights and applications. Each of the principle research areas are briefly summarized next. Mixed type random variables are a hybrid of continuous and discrete random variables, having components of both continuous probability density and discrete probability mass. This dissertation illustrates the challenges inherent in plotting mixed type distributions, and introduces an algorithm that addresses those issues. It considers sums and products of mixed type random variables, and supports its conclusions using Monte Carlo simulation experiments. Lastly, it introduces MixedAPPL, a computer algebra system software package designed for manipulating mixed type random variables. Confidence regions are a multi-dimensional version of a confidence interval. They are helpful to visualize and quantify uncertainty surrounding a point estimate. We begin by developing efficient plot algorithms for two-dimensional confidence regions. This research focuses specifically on likelihood-ratio based confidence regions for two-parameter univariate probability models, although the plot techniques are transferable to any two-dimensional setting. The R package 'conf' is introduced, which automates these confidence region plot algorithms for complete and right-censored data sets. Among its benefits, 'conf' provides access to Monte Carlo simulation experiments for confidence region coverage to an extent not possible previously. The corresponding coverage analysis results include reference tables for the Weibull, normal, and log-logistic distributions. These reference tables yield confidence region plots with exact coverage. The final topic is the introduction and analysis of a golden quantile rank set (GQRS). The term quantile rank set is used here to denote the population cumulative distribution function values corresponding to a sample. A GQRS can be thought of as "perfectly" representative of their population distribution because samples corresponding to a GQRS result in an estimator(s) matching the associated true population parameter(s). This unique characteristic is not applicable for all estimators and/or distributions, but when present, provides valuable insights and applications. Specifically, applications include an alternative (and at times computationally superior) method for parameter estimation and an exact actual coverage methodology for confidence regions (at times in which currently only estimates exist). Distributions with a GQRS associated with maximum likelihood estimation include the normal, exponential, Weibull, log logistic, and one-parameter exponential power distributions.

Arab Muslim immigrant women encounter many stressors and are at risk for depression. Social supports from husbands, family and friends are generally considered mitigating resources for depression. However, changes in social support over time and the effects of such supports on depression at a future time period have not been fully addressed in the literature This thesis investigated the relationship between demographic characteristics, changes in social support, and depression in Arab Muslim immigrant women to the USA. A sample of 454 married Arab Muslim immigrant women provided demographic data, scores on social support variables and depression at three time periods approximately six months apart. Various statistical techniques at our disposal such as boxplots, response curves, descriptive statistics, ANOVA and ANCOVA, simple and multiple linear regressions have been used to see how various factors and variables are associated with changes in social support from husband, extended family and friend over time. Simple and multiple regression analyses are carried out to see if any variable observed at the time of first survey can be used to predict depression at a future time. Social support from husband and friend, husband's employment status and education, and depression at time one are found to be significantly associated with depression at time three. Finally, logistic regression analysis conducted for a binary depression outcome variable indicated that lower total social support and higher depression score of survey participants at the time of first survey increase their probability of being depressed at the time of third survey.

In this thesis, the basics of smoothing spline analysis of variance are first introduced. Regular physical activity has been shown to reduce the risk of chronic diseases in older adults, such as heart disease, stroke, diabetes, and certain forms of cancer. Accurate measurement of physical activity levels in older adults is crucial to identify those who may require interventions to increase their activity levels and prevent functional decline. In our study, we collected data on the physical activity of older individuals by utilizing accelerometer accelerometers. To estimate the underlying patterns related to each covariate, we applies smoothing spline analysis of variance (SSANOVA) methods to two types of measurements from the accelerometer device. We investigates the underlying patterns of different participant groups and compared the patterns among groups. The paper reveals clear patterns of activity levels throughout the day and across days, with differences among groups observed. Additionally, the study compares the mean curve method and the SSANOVA model, and shows that the SSANOVA model is a more suitable method for analyzing physical activity data. The study provides valuable insights into daily physical activity patterns in older people and highlights the usefulness of the SSANOVA model for such data analysis.