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The New Jersey Institute of Technology's
Electronic Theses & Dissertations Project

Title: Modeling with bivariate geometric distributions
Author: Li, Jing
View Online: njit-etd2010-097
(ix, 55 pages ~ 2.0 MB pdf)
Department: Department of Mathematical Sciences
Degree: Doctor of Philosophy
Program: Mathematical Sciences
Document Type: Dissertation
Advisory Committee: Dhar, Sunil Kumar (Committee chair)
Bhattacharjee, Manish Chandra (Committee member)
Subramanian, Sundarraman (Committee member)
Guo, Wenge (Committee member)
Jain, Aridaman Kumar (Committee member)
Subramanian, Ganesh (Committee member)
Date: 2010-05
Keywords: Bivariate geometric distribution
Conditional failure rate
Maximum likelihood estimation
Bayes estimation
Availability: Unrestricted

This dissertation studied systems with several components which were subject to different types of failures. Systems with two components having frequency counts in the domain of positive integers, and the survival time of each component following geometric or mixture geometric distribution can be classified into this category. Examples of such systems include twin engines of an airplane and the paired organs in a human body. It was found that such a system, using conditional arguments, can be characterized as multivariate geometric distributions. It was proved that these characterizations of the geometric models can be achieved using conditional probabilities, conditional failure rates, or probability generating functions. These new models were fitted to real-life data using the maximum likelihood estimators, Bayes estimators, and method of moment estimators. The maximum likelihood estimators were obtained by solving score equations. Two methods of moments estimators were compared in each of the several bivariate geometric models using the estimated bias vectors and the estimated variance-covariance matrices. This comparison was done through a Monte-Carlo simulation for increasing sample sizes. The Chi-square goodness-of-fit tests were used to evaluate model performance.

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