Nonparametric bootstrapping of the reliability function for multiple copies of a repairable item modeled by a birth process

Nonparametric bootstrap inference is developed for the reliability function estimated from censored, nonstationary failure time data for multiple copies of repairable items. We assume that each copy has a known, but not necessarily the same, observation period; and upon failure of one copy, design modifications are implemented for all copies operating at that time to prevent further failures arising from the same fault. This implies that, at any point in time, all operating copies will contain the same set of faults. Failures are modeled as a birth process because there is a reduction in the rate of occurrence at each failure. The data structure comprises a mix of deterministic and random censoring mechanisms corresponding to the known observation period of the copy, and the random censoring time of each fault. Hence, bootstrap confidence intervals and regions for the reliability function measure the length of time a fault can remain within the item until realization as failure in one of the copies. Explicit formulae derived for the re-sampling probabilities greatly reduce dependency on Monte-Carlo simulation. Investigations show a small bias arising in re-sampling that can be quantified and corrected. The variability generated by the re-sampling approach approximates the variability in the underlying birth process, and so supports appropriate inference. An illustrative example describes application to a problem, and discusses the validity of modeling assumptions within industrial practice