On the reliability of Type II censored reliability analyses. - See Ju Chua (2009)
This thesis considers the analysis of reliability data subject to censoring, and, in particular, the extent to which an interim analysis - here, using information based on Type II censoring - provides a guide to the final analysis. Under a Type II censored sampling, a random sample of n units is put on test simultaneously, and the test is terminated as soon as r (1 ≤ r ≤ n, although we are usually interested in r < n) failures are observed. In the case where all test units were observed to fail (r = n), the sample is complete. From a statistical perspective, the analysis of the complete sample is to be preferred, but, in practice, censoring is often necessary; such sampling plan can save money and time, since it could take a very long time for all units to fail in some instances. From a practical perspective, an experimenter may be interested to know the smallest number of failures at which the experiment can be reasonably or safely terminated with the interim analysis still providing a close and reliable guide to the analysis of the final, complete data. In this thesis, we aim to gain more insight into the roles of censoring number r and sample size n under this sampling plan. Our approach requires a method to measure the precision of a Type II censored estimate, calculated at censoring level r, in estimating the complete estimate, and hence the study of the relationship between interim and final estimates. For simplicity, we assume that the lifetimes follow the exponential distribution, and then adopt the methods to the two- parameter Weibull and Burr Type XII distributions, both are widely used in reliability modelling. We start by presenting some mathematical and computational methodology for estimating model parameters and percentile functions, by the method of maximum likelihood. Expressions for the asymptotic variances and covariances of the estimators are given. In practice, some indication of the likely accuracy of these estimates is often desired; the theory of asymptotic Normality of maximum likelihood estimator is convenient, however, we consider the use of relative likelihood contour plots to obtain approximate confidence regions of parameters in relatively small samples. Finally, we provide formulae of the correlations between the interim and final maximum likelihood estimators of model parameters and a particular percentile function, and discuss some practical implications of our work, based on the results obtained from published data and simulation experiments.