Graduation Year

2011

Document Type

Dissertation

Degree

Ph.D.

Degree Granting Department

Mathematics and Statistics

Major Professor

Christos P. Tsokos, Ph.D.

Committee Member

Marcus McWaters, Ph.D.

Committee Member

Kandethody M. Ramachandran, Ph.D.

Committee Member

Rebecca D. Wooten, Ph.D.

Keywords

Ordinary Bayes, Power Law, Loss function, Weibull process, Breast cancer

Abstract

The objective of this study is to compare Bayesian and parametric approaches to determine the best for estimating reliability in complex systems. Determining reliability is particularly important in business and medical contexts. As expected, the Bayesian method showed the best results in assessing the reliability of systems.

In the first study, the Bayesian reliability function under the Higgins-Tsokos loss function using Jeffreys as its prior performs similarly as when the Bayesian reliability function is based on the squared-error loss. In addition, the Higgins-Tsokos loss function was found to be as robust as the squared-error loss function and slightly more efficient.

In the second study, we illustrated that--through the power law intensity function--Bayesian analysis is applicable in the power law process. The power law intensity function is the key entity of the power law process (also called the Weibull process or the non-homogeneous Poisson process). It gives the rate of change of a system's reliability as a function of time. First, using real data, we demonstrated that one of our two parameters behaves as a random variable. With the generated estimates, we obtained a probability density function that characterizes the behavior of this random variable. Using this information, under the commonly used squared-error loss function and with a proposed adjusted estimate for the second parameter, we obtained a Bayesian reliability estimate of the failure probability distribution that is characterized by the power law process. Then, using a Monte Carlo simulation, we showed the superiority of the Bayesian estimate compared with the maximum likelihood estimate and also the better performance of the proposed estimate compared with its maximum likelihood counterpart.

In the next study, a Bayesian sensitivity analysis was performed via Monte Carlo simulation, using the same parameter as in the previous study and under the commonly used squared-error loss function, using mean square error comparison. The analysis was extended to the second parameter as a function of the first, based on the relationship between their maximum likelihood estimates. The simulation procedure demonstrated that the Bayesian estimates are superior to the maximum likelihood estimates and that the selection of the prior distribution was sensitive. Secondly, we found that the proposed adjusted estimate for the second parameter has better performance under a noninformative prior.

In the fourth study, a Bayesian approach was applied to real data from breast cancer research. The purpose of the study was to investigate the applicability of a Bayesian analysis to survival time of breast cancer data and to justify the applicability of the Bayesian approach to this domain. The estimation of one parameter, the survival function, and hazard function were analyzed. The simulation analysis showed that the Bayesian estimate of the parameter performed better compared with the estimated value under the Wheeler procedure. The excellent performance of the Bayesian estimate is reflected even for small sample sizes. The Bayesian survival function was also found to be more efficient than its parametric counterpart.

In the last study, a Bayesian analysis was carried out to investigate the sensitivity to the choice of the loss function. One of the parameters of the distribution that characterized the survival times for breast cancer data was estimated applying a Bayesian approach and under two different loss functions. Also, the estimates of the survival function were determined under the same setting. The simulation analysis showed that the choice of the squared-error loss function is robust in estimating the parameter and the survival function.

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