In this thesis we present two approximations to the distribution function of the sum of n independent random variables. They are obtained from generalizations of asymptotic expansions derived by Rubin and Zidek who considered the case of chi random variables. These expansions are obtained from Gurland's inversion formula for the distribution function by using an adaptation of Laplace's method for integrals. By means of numerical results obtained for a variety of common distributions and small values of n these approximations arc compared to the classical methods of Edgeworth and Cramer. Finally, the method is used to obtain approximations to the non-central chi-square distribution and to the doubly non-central F-distribution for various cases defined in terms of its parameters. / Science, Faculty of / Mathematics, Department of / Graduate

During the past three decades or so there has been much work done concerning contagious probability distributions in an attempt to explain the behavior of certain types of biological populations. The distributions most widely discussed have been the Poisson-binomial, the Poisson Pascal or Poisson-negative binomial, and the Poisson-Poisson or Neyman Type A. Many generalizations of the above distributions have also been discussed. The purpose of this work is to discuss the multivariate analogues of the above three distributions, i.e. the Poisson-multinomial, Poisson-negative multinomial, and Poisson-multivariate Poisson, respectively. In chapter one the first of these distributions is discussed. Initially a biological model is suggested which leads us to a probability generating function. From this a recursion formula for the probabilities is found. Parameter estimation by the methods of moments and maximum likelihood is discussed in some detail and an approximation for the asymptotic efficiency of the former method is found. The latter method is asymptotically efficient. Finally sample zero and unit sample frequency estimators are briefly discussed. In chapter two, exactly the same procedure is followed for the Poisson-negative multinomial distribution. Many close similarities are obvious between the two distributions. The last chapter is devoted to a particular common limiting case of the first two distributions. This is the Poisson-multivariate Poisson. In this case the desired results are obtained by carefully considering appropriate limits in either of the previous two cases. / Science, Faculty of / Mathematics, Department of / Graduate

Let ℱ = {F} be a family of n-variate cumulative distribution functions (c.d.f.'s). If F₁...,F(k) belong to ℱ and P₁,...,P(k) are positive numbers that sum to 1, then the convex combination M(x₁ ,. . . , x(n)) = [formula omitted](x₁,...,x(n)) is called a finite mixture generated by ℱ. The F₁,…,F(k) are called the components of the mixture and the P₁,…,P(k) are called their weights, respectively. The mixture M(x₁,...,x(n)) is said to be identifiable with respect to ℱ if no other convex combination of a finite number of c.d.f.'s from ℱ will generate M(x₁,...,x(n)). We establish the identiflability of mixtures consisting of at most k components when the components belong to a family of univariate c.d.f.'s that have the following properties: (a) no two c.d.f.'s have the same mean; (b) each c.d.f. has the same r(th) central moment for r = 1,...,2k-1; and (c) the first 2k-1 central moments are finite. If the mixture and the 2k-1 central moments are known, a solution for the weights and means of the components is given. If a random sample is taken from the mixture, then asymptotically normal estimates of the weights and means are given, providing the 2k-1 central moments are known. Matrix mixtures are introduced and are found to be of use in estimating the density functions and c.d.f.'s of the components. In the .case of the above family, the estimates of the density functions are shown to have an asymptotically normal distribution. Consistent and least squares estimates are obtained for the component c.d.f.'s. We show that for multivariate mixtures identifiability of any one of the marginal mixtures implies the identifiability of the multivariate mixture, but not conversely. Finally, the univariate results are generalized to the multivariate case, and an example of the use of matrix mixtures is given. / Science, Faculty of / Mathematics, Department of / Graduate

The work reviews a article published by N. T. Feather about five approaches which relate to the analysis of behavior in a choice situation where a decision is made between alternatives having different subjective probabilities of attainment. The relationship between choice potential and success probability is affect by the type of situation in which the choice is made.

Customer Lifetime Value (CLV) ---which is a measure of the profit generating potential, or value, of a customer---is increasingly being considered a touchstone for customer relationship management. As the guide and benchmark for Customer Relationship Management (CRM) applications, CLV analysis has received increasing attention from both the marketing practitioners and researchers from different domains. Furthermore, the central challenge in predicting CLV is the precise calculation of customer’s length of service (LOS). There are several statistical approaches for this problem and several researchers have used these approaches to perform survival analysis in different domains. However, classical survival analysis techniques like Kaplan-Meier approach which offers a fully non-parametric estimate ignores the covariates completely and assumes stationary of churn behavior along time, which makes it less practical. Further, segments of customers, whose lifetimes and covariate effects can vary widely, are not necessarily easy to detect. Like many other applications, data mining is emerging as a compelling analysis tool for the CLV application recently. Comparatively, data mining methods offer an interesting alternative with the fact that they are less limited than the conventional statistical approaches. Customer databases contain histories of vital events such as the acquisition and cancellation of products and services. The historical data is used to build predictive models for customer retention, cross-selling, and other database marketing endeavors. In this research project we discuss and investigate the possibility of combining these statistical approaches with data mining methods to improve the performance for the CLV problem in a real business context. Part of the research effort is placed on the precise prediction of LOS of the customers in concentration of a real world business. Using the conventional statistical approaches and data mining methods in tandem, we demonstrate how data mining tools can be apt complements of the classical statistical models ---resulting in a CLV prediction model that is both accurate and understandable. We also evaluate the proposed integrated method to extract interesting business domain knowledge within the scope of CLV problem. In particular, several data mining methods are discussed and evaluated according to their accuracy of prediction and interpretability of results. The research findings will lead us to a data mining method combined with survival analysis approaches as a robust tool for modeling CLV and for assisting management decision-making. A calling plan strategy is designed based on the predicted survival time and calculated CLV for the telecommunication industry. The calling plan strategy further investigates potential business knowledge assisted by the CLV calculated.

There are several solutions for analysis of clustered binary data. However, the two most common tools in use today, generalized estimating equations and random effects or mixed models, rely heavily on asymptotic theory. However, in many situations, such as small or sparse samples, asymptotic assumptions may not be met. For this reason we explore the utility of the quadratic exponential model and conditional analysis to estimate the effect size of a trend parameter in small sample and sparse data settings. Further we explore the computational efficiency of two methods for conducting conditional analysis, the network algorithm and Markov chain Monte Carlo. Our findings indicate that conditional estimates do indeed outperform their unconditional maximum likelihood counterparts. The network algorithm remains the fastest tool for generating the required conditional distribution. However, for large samples, the Markov chain Monte Carlo approach accurately estimates the conditional distribution and is more efficient than the network algorithm.

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A probability programming language is developed and presented; applications illustrate its use. Algorithms and generalized theorems used in probability are encapsulated into a programming environment with the computer algebra system Maple to provide the applied community with automated probability capabilities. Algorithms of procedures are presented and explained, including detailed presentations on three of the most significant procedures. Applications that encompass a wide range of applied topics including goodness-of-fit testing, probabilistic modeling, central limit theorem augmentation, generation of mathematical resources, and estimation are presented.

Carbonate strata preserve indicators of local marine environments through time. Such indicators often include microfossils that have relatively unique conditions under which they can survive, including light, nutrients, salinity, and especially water temperature. As such, microfossils are environmental proxies. When these microfossils are preserved in the rock record, they constitute key components of depositional facies. Spence et al. (2004, 2007) has proposed several approaches for determining the facies of a given stratigraphic succession based upon these proxies. Cluster analysis can be used to determine microfossil groups that represent specific environmental conditions. Identifying which microfossil groups exist through time can indicate local environmental change. When new observations (microfossils) are found, classification analysis can be used to predict group membership. Kristen Briggs (2005) identified the microfossils present in sedimentary strata within a specific time interval (Morrowan) of Pennsylvanian-age rocks. In this study we expand analysis to overlying Atokan and Desmoinesian strata. The Bird Spring Formation in Arrow Canyon, Nevada records cycles of environmental change as evidenced by changes in microfossils. Our research investigates cluster and classification analyses as tools for determining the marine facies succession. Light, nutrients, salinity, and water temperature are very dependent on water depth; therefore, our analyses essentially indicate the relative rise and fall of sea-level during Early to Middle Pennsylvanian time.