Bayesian Approaches to Parameter Estimation of Load-Sharing and Competing Risks Systems

Kostan, Travis
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In many engineering applications, researchers are commonly interested in making statistical inference about the reliability of certain items or individuals, such as the strength of fibrous materials. Reliability analysis often requires accurate estimation of unknown model parameters from multiple lifetime distributions. In this dissertation, we primarily focus on two popular systems; the parallel load-sharing system and the competing risks system, which arise often in the fields of manufacturing engineering and medical studies. We develop Bayesian parameter estimation methods for these systems and demonstrate its robustness to changes in the underlying lifetime distribution choices and prior hyperparameter specifications. First, for the parallel load-sharing system, by assuming that the lifetimes of the components follow a Weibull distribution, we develop objective Jeffreys priors for the unknown model parameters, and we prove the propriety of the resulting joint posterior distribution. To demonstrate the benefits of incorporating prior information into the Bayesian estimation, we additionally develop the joint posterior distribution assuming informative gamma priors. Then, we develop an efficient and robust generalized ratio-of-uniforms sampling technique to generate posterior samples from the joint posterior distributions and allow for posterior inference such as Bayesian estimates and credible intervals of the unknown model parameters. Unlike traditional Markov chain Monte Carlo (MCMC) methods, the proposed sampling algorithm can efficiently generate independent posterior samples from an incognizable posterior distribution. Numerical results from simulation studies and a real-data application show that the proposed Bayesian parameter estimation methodology is superior to existing methods for both large and small sample sizes, regardless of whether the considered priors are informative or non-informative. We show that our results using the generalized ratio-of-uniforms posterior sampling are easily extendable to other distributions by also considering both informative and non-informative priors when the components' lifetimes follow the Lindley distribution, and we provide further simulations to support the accuracy and reliability of our approach. Finally, for the competing risks system, we add to the existing research by showing that the proposed Bayesian approach including the novel generalized ratio-of-uniforms posterior sampling is a viable alternative to traditional approaches. We derive Jeffreys prior for the unknown parameters if the competing risks follow an exponential distribution, and again provide proof of propriety of the resulting joint posterior distribution. For making draws from the joint posterior distribution, we propose a novel sampling algorithm combining the generalized ratio-of-uniforms method with Gibbs sampling. We conduct simulation studies to show that the proposed method provides excellent results even in more complicated cases where potential exists for both censored data and for missing or unknown causes of failure.

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Bayesian, Ratio-of-uniforms, Markov chain Monte Carlo, Unknown model parameters
Management Science and Statistics