<p>In this thesis, inferential methods for one-shot device testing data from accelerated life-test are developed. Due to constraints on time and budget, accelerated life-tests are commonly used to induce more failures within a reasonable amount of test-time for obtaining more lifetime information that will be especially useful in reliability analysis. One-shot devices, which can be used only once as they get destroyed immediately after testing, yield observations only on their condition and not on their real lifetimes. So, only binary response data are observed from an one-shot device testing experiment. Since no failure times of units are observed, we use the EM algorithm for determining the maximum likelihood estimates of the model parameters. Also, inference for the reliability at a mission time and the mean lifetime at normal operating conditions are also developed.</p> <p>The thesis proceeds as follows. Chapter 2 considers the exponential distribution with single-stress relationship and develops inferential methods for the model parameters, the reliability and the mean lifetime. The results obtained by the EM algorithm are compared with those obtained from the Bayesian approach. A one-shot device testing data is analyzed by the proposed method and presented as an illustrative example. Next, in Chapter 3, the exponential distribution with multiple-stress relationship is considered and corresponding inferential results are developed. Jackknife technique is described for the bias reduction in the developed estimates. Interval estimation for the reliability and the mean lifetime are also discussed based on observed information matrix, jackknife technique, parametric bootstrap method, and transformation technique. Again, we present an example to illustrate all the inferential methods developed in this chapter. Chapter 4 considers the point and interval estimation for the one-shot device testing data under the Weibull distribution with multiple-stress relationship and illustrates the application of the proposed methods in a study involving the development of tumors in mice with respect to risk factors such as sex, strain of offspring, and dose effects of benzidine dihydrochloride. A Monte Carlo simulation study is also carried out to evaluate the performance of the EM estimates for different levels of reliability and different sample sizes. Chapter 5 describes a general algorithm for the determination of the optimal design of an accelerated life-test plan for one-shot device testing experiment. It is based on the asymptotic variance of the estimated reliability at a specific mission time. A numerical example is presented to illustrate the application of the algorithm. Finally, Chapter 6 presents some concluding remarks and some additional research problems that would be of interest for further study.</p> / Doctor of Philosophy (PhD)

In this thesis, we develop some inferential results for one-shot device testing data analysis. These extend and generalize existing methods in the literature. First, a competing-risk model is introduced for one-shot testing data under accelerated life-tests. One-shot devices are products which will be destroyed immediately after use. Therefore, we can observe only a binary status as data, success or failure, of such products instead of its lifetime. Many one-shot devices contain multiple components and failure of any one of them will lead to the failure of the device. Failed devices are inspected to identify the specific cause of failure. Since the exact lifetime is not observed, EM algorithm becomes a natural tool to obtain the maximum likelihood estimates of the model parameters. Here, we develop the EM algorithm for competing exponential and Weibull cases. Second, a semi-parametric approach is developed for simple one-shot device testing data. Semi-parametric estimation is a model that consists of parametric and non-parametric components. For this purpose, we only assume the hazards at different stress levels are proportional to each other, but no distributional assumption is made on the lifetimes. This provides a greater flexibility in model fitting and enables us to examine the relationship between the reliability of devices and the stress factors. Third, Bayesian inference is developed for one-shot device testing data under exponential distribution and Weibull distribution with non-constant shape parameters for competing risks. Bayesian framework provides statistical inference from another perspective. It assumes the model parameters to be random and then improves the inference by incorporating expert's experience as prior information. This method is shown to be very useful if we have limited failure observation wherein the maximum likelihood estimator may not exist. The thesis proceeds as follows. In Chapter 2, we assume the one-shot devices to have two components with lifetimes having exponential distributions with multiple stress factors. We then develop an EM algorithm for developing likelihood inference for the model parameters as well as some useful reliability characteristics. In Chapter 3, we generalize to the situation when lifetimes follow a Weibull distribution with non-constant shape parameters. In Chapter 4, we propose a semi-parametric model for simple one-shot device test data based on proportional hazards model and develop associated inferential results. In Chapter 5, we consider the competing risk model with exponential lifetimes and develop inference by adopting the Bayesian approach. In Chapter 6, we generalize these results on Bayesian inference to the situation when the lifetimes have a Weibull distribution. Finally, we provide some concluding remarks and indicate some future research directions in Chapter 7. / Thesis / Doctor of Philosophy (PhD)

A device that is accompanied by an irreversible chemical reaction or physical destruction and could no longer function properly after performing its intended function is referred to as a one-shot device. One-shot device test data differ from typical data obtained by measuring lifetimes in standard life-tests. Due to the very nature of one-shot devices, actual lifetimes of one-shot devices under test cannot be observed, and they are either left- or right-censored. In addition, a one-shot device often has multiple components that could cause the failure of the device. The components are coupled together in the manufacturing process or assembly, resulting in the failure modes possessing latent heterogeneity and dependence. Frailty models enable us to describe the influence of common, but unobservable covariates, on the hazard function as a random effect in a model and also provide an easily understandable interpretation. In this thesis, we develop some inferential results for one-shot device testing data with gamma frailty model. We first develop an efficient expectation-maximization (EM) algorithm for determining the maximum likelihood estimates of model parameters of a gamma frailty model with exponential lifetime distributions for components based on one-shot device test data with multiple failure modes, wherein the data are obtained from a constant-stress accelerated life-test. The maximum likelihood estimate of the mean lifetime of $k$-out-of-$M$ structured one-shot devices under normal operating conditions is also presented. In addition, the asymptotic variance–covariance matrix of the maximum likelihood estimates is derived, which is then used to construct asymptotic confidence intervals for the model parameters. The performance of the proposed inferential methods is finally evaluated through Monte Carlo simulations and then illustrated with a numerical example. A gamma frailty model with Weibull baseline hazards is considered next for fitting one-shot device testing data. The Weibull baseline hazards enable us to analyze time-varying failure rates more accurately, allowing for a deeper understanding of the dynamic nature of system's reliability. We develop an EM algorithm for estimating the model parameters utilizing the complete likelihood function. A detailed simulation study evaluates the performance of the Weibull baseline hazard model with that of the exponential baseline hazard model. The introduction of shape parameters in the component's lifetime distribution within the Weibull baseline hazard model offers enhanced flexibility in model fitting. Finally, Bayesian inference is then developed for the gamma frailty model with exponential baseline hazard for one-shot device testing data. We introduce the Bayesian estimation procedure using Markov chain Monte Carlo (MCMC) technique for estimating the model parameters as well as for developing credible intervals for those parameters. The performance of the proposed method is evaluated in a simulation study. Model comparison between independence model and the frailty model is made using Bayesian model selection criterion. / Thesis / Candidate in Philosophy