Parametric Bootstrap Confidence Interval Method for the Power Law Process With Applications to Multiple Repairable Systems

In the reliability analysis of repairable systems, the power law process (PLP) is most commonly used to model the failure process of a repairable system. The estimates of some important functions of the PLP parameters have played an important role in assessing systems reliability or making reliability management decisions. For the general cases where multiple identical repairable systems operate independently for different periods of time and failures for each system following the same PLP, this paper proposes a parametric bootstrap method to construct highly accurate confidence intervals on any function of the PLP parameters. The proposed method is based on the log-ratios transformation. This transformation with the nice feature of being independent of the PLP parameters is applied to the failure times of multiple systems to obtain a random sample from an exponential distribution. Based on this sample and the relationship between the parameter of the exponential distribution and the PLP parameters, the confidence intervals on the function of the PLP parameters can be easily constructed by three bootstrap confidence interval methods. Numerical experiments illustrate the rationality and validity of the proposed confidence interval method with applications to improving systems, deteriorating systems, and systems with constant failure rate, in contrast to Crow's intervals and asymptotic intervals. Summarizing the applications of the proposed method to several functions of the PLP parameters of practical interests, we provide some suggestions on the selections of the three bootstrap confidence interval methods. Finally, two real examples are presented.