A Method of Applying an Unbiased Estimator for Monitoring the Weibull Shape Parameter and Time-to-Event Model Using Copula with Accelerated Life Testing
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Two topics are discussed in this dissertation. In chapter 1, we introduce a method for monitoring the Weibull shape parameter beta without specifying the values of nuisance parameters. The control limits depend on the sample size, target average run length, and the stable value of beta. The only assumption on the nuisance parameter is that the scale parameter alpha remains unchanged during each sampling period. The proposed methodutilizes the relationship between Weibull and smallest extreme value distribution. By calculating the absolute deviations from the largest sample value, an unbiased control chart for beta can be constructed. The control limit for one-sided and two-sided chart has been derived for several all-OK average run length (ARL) in this paper. Two schemes are discussed: the control-limits-only scheme, and the control-limits-with-warning-lines scheme. The ARL performance is studied and compared to other two methods to monitorbeta under similar assumptions.In chapter 2, we introduce a copula based model for time-to-event data. Time-to-event data is one of the most common type of data in many areas, especially, engineering and medical field. Competing risks and semi-competing risks data can be considered as special type of time-to-event data in which several observations are censored, and hence, unobservable. Due to the complexity of modern electronic products, the lifespan of a device can be suppressed by one of several components and very often, lifetime of those individual components are not independent on each other. In our method, the dependence is modeled by using a copula function. In this article, we introduce a full parametric statistical method for fitting multivariate censored time-to-event data using copula. This method utilized the information in the form of combinations of exact failure and right censored observation under multivariate data structure. Goodness-of-fit diagnostics in comparing model fits to data are performed via quantile-quantile plots, and the asymptotic properties are studied via numerical simulations. The model is applied to an accelerated lifetime testing dataset on insulation with three failure modes.