In recent actuarial literature, the bivariate Poisson regression model has been found to be useful for modeling paired count data. However, the basic assumption of marginal equi-dispersion may be quite restrictive in practice. To overcome this limitation, we consider here the recently developed bivariate Conway–Maxwell–Poisson (CMP) distribution. As a distribution that allows data dispersion, the bivariate CMP distribution is a flexible distribution which includes the bivariate Poisson, bivariate Bernoulli and bivariate Geometric distributions all as special cases. We discuss inferential methods for this CMP distribution. An application to automobile insurance data demonstrates its usefulness as an alternative framework to the commonly used bivariate Poisson model. / Thesis / Master of Science (MSc)

No description available.

Same types of systems with the same age usually have different amounts of cumulated usage. These systems when in operation usually have different performance and effectiveness. In this case the existing models of the univariate measures of system effectiveness are inadequate and incomplete. For example, the univariate availability measures for these same-aged systems are all the same even though with different amounts of usage. This is the motivation for this research to pursue a bivariate approach in reliability and maintenance modeling. This research presents a framework for bivariate modeling of a single-unit system. Five key efforts are identified and organized as: (i) bivariate failure modeling, (ii) bivariate renewal modeling, (iii) bivariate corrective maintenance (CM) modeling, (iv) bivariate preventive maintenance (PM) modeling, and (v) bivariate availability modeling. The results provide a foundation for further study of bivariate and multivariate models. For bivariate failure modeling, several bivariate failure models are constructed to represent the possible correlation structures of the two system aging variables, time and usage. The behavior of these models is examined under various correlation structures. The developed models are used to analyze example maintenance problems. Models for bivariate renewal, bivariate CM, and bivariate PM are derived based on the constructed bivariate failure models and the developed bivariate renewal theory. For bivariate CM modeling, corrective maintenance is modeled as an alternating bivariate renewal process or simply an ordinary bivariate renewal process. For bivariate PM modeling, PM models are examined under a bivariate age replacement preventive maintenance policy. The Laplace transforms of the renewal functions (and densities) for these models are obtained. Definitions for bivariate availability functions are developed. Based on the derived CM and PM models, the Laplace transforms for their corresponding bivariate availability models are constructed. The idea of the quality of availability measure is also defined in terms of bivariate availability models. The most significant observation is that this framework provides a new way to study the reliability and maintenance of equipment for which univariate measures are incomplete. Therefore, a new area of reliability research is identified. The definitions offered may be modified and the approach to model formulation presented may be used to define other models. / Ph. D.

<p> This thesis deals with order statistics and the asymptotic distributions of certain functions of the first and second quartiles of a sample drawn at random from a bivariate population, whose distribution function is specified by its truncated bivariate Edgeworth Series.</p> / Thesis / Master of Science (MSc)

No description available.

Equipment behavior is often discussed in terms of age and use. For example, an automobile is frequently referred to 3 years old with 30,000 miles. Bivariate failure modeling provides a framework for studying system behavior as a function of two variables. This is meaningful when studying the reliability/availability of systems and equipment. This thesis extends work done in the area of bivariate failure modeling. Four bivariate failure models are selected for analysis. The study includes exploration of bivariate random number generation. The random data is utilized in estimating the bivariate renewal function and bivariate availability function. The two measures provide insight on system behavior characterized by multiple variables. A method for generating bivariate failure and repair data is developed for each model. Of the four models, two represent correlated random variables; the other two, stochastic functionally dependent variables. Also, methods of estimating the bivariate renewals function and bivariate availability function are constructed. The bivariate failure and repair data from the four failure models is incorporated into the estimation processes to study various failure scenarios. / Master of Science

Obesity is a growing public health concern that is being confronted by both developed and developing countries. South Africa is no exception, facing the highest burden of obesity amongst African countries. Using two waves of data from the National Income Dynamics Study, this study aims to investigate the relationship between obesity and employment status for working age individuals in the context of the South African labour market. This study contributes to existing literature on this subject by explicitly accounting for potential simultaneity and endogeneity between obesity and employment. Given the hypothesised two-way causal relationship between obesity and unemployment, two different models are used to assess whether this issue exists for the dataset; a bivariate probit model to assess if there is a bivariate relationship between obesity and employment, and a recursive bivariate probit model to assess if obesity is an endogenous regressor of employment. A change in state univariate probit model is then implemented across the two waves to better understand if fluctuations in weight status are a result of labour market state transitions. The results of the study show that obesity and employment are independent in the bivariate probit models and obesity is an exogenous regressor of employment status in the recursive bivariate probit models. Changes in labour market state do not have a significant impact on the probability of transitioning to obese compared to no changes in labour market state, bar transitioning from not economically active to employed, which increases the probability of becoming obese. The findings suggest that, in the South African labour market context, obesity and employment are not related, indicating that there are other underlying factors, such as nutritional intake and genetic composition, that may contribute to fluctuations in weight status. The results suggest that obesity is prolific in South Africa, and impacts individuals across the entire distribution for labour market status and income.

This thesis examines the labor force participation decisions of married couples, and special attention is paid to a spouse’s health conditions affecting their own and the spouse’s labor force participation decision. I used the Health and Retirement Study survey data and estimated a seemingly unrelated bivariate probit model. A number of variables besides health condition were added: age, education level, and family unearned income. The results of this research paper support the findings from the relevant literature that the labor supply decisions of the husband and wife are related. The oldest age group is least likely to work. The younger the husband, the more likely it is that the husband will work. At the ages between 40 and 49, wives have the biggest probability to work. The higher the education level, the more likely it is that a spouse is going to work. The more total family unearned income, the less probable the spouse will go to work. Poor health has a negative effect on labor force participation and a positive effect for the spouse’s labor force participation.

In this thesis, we present a method for the automated generation of numerical evaluation routines for bivariate functions via tensor product series and develop a toolkit to assist with the generation of the approximations. The final approximations can be evaluated in user-defined precision or in hardware floating point precision by default. The evaluation routines can also be compiled into a C library (or a library in some other language) for more efficient evaluations. The toolkit can be used for various mathematical functions of two variables, such as Bessel functions or user-defined functions, at any given precision. The method of tensor product series expansion reduces the bivariate approximation problem to a sequence of univariate approximation problems. In order to control the degrees of the approximating functions so that evaluation will be accurate and efficient, we recursively divide the bivariate intervals into subintervals until both the number of terms in the tensor product series and the degrees of the univariate approximations are less than specified bounds. We then generate in each subinterval rational approximations using Chebyshev-Padé approximants or polynomial approximations using Chebyshev series according to the user's specification. Finally we show the experimental results for a variety of bivariate functions, which achieve a significant speedup over the original Maple functions for evaluation in hardware floating point precision. We also compare the results of choosing polynomial approximations versus rational approximations for the univariate subproblems.

In this thesis, we present a method for the automated generation of numerical evaluation routines for bivariate functions via tensor product series and develop a toolkit to assist with the generation of the approximations. The final approximations can be evaluated in user-defined precision or in hardware floating point precision by default. The evaluation routines can also be compiled into a C library (or a library in some other language) for more efficient evaluations. The toolkit can be used for various mathematical functions of two variables, such as Bessel functions or user-defined functions, at any given precision. The method of tensor product series expansion reduces the bivariate approximation problem to a sequence of univariate approximation problems. In order to control the degrees of the approximating functions so that evaluation will be accurate and efficient, we recursively divide the bivariate intervals into subintervals until both the number of terms in the tensor product series and the degrees of the univariate approximations are less than specified bounds. We then generate in each subinterval rational approximations using Chebyshev-Padé approximants or polynomial approximations using Chebyshev series according to the user's specification. Finally we show the experimental results for a variety of bivariate functions, which achieve a significant speedup over the original Maple functions for evaluation in hardware floating point precision. We also compare the results of choosing polynomial approximations versus rational approximations for the univariate subproblems.