Inactivity times of components upon system failure with application to missing data problems

We introduce a concept of inactivity times of components upon system failure, i.e., times that elapsed until the breakdown of the system from moments of failures of its non-surviving components. These times are related to autopsy data and are useful, for instance, for predicting exact instants of component failures. Under the assumption that the joint distribution of component lifetimes is exchangeable and absolutely continuous, we obtain a formula for the joint survival function of the inactivity times of interest. We express this formula in terms of the copula modelling the dependence structure among components. To analyze how the dependence among the components and other factors affect the studied inactivity times, we demonstrate numerical results for some special cases with specific copulas and margins. Finally, we give an example of an application of the presented results to the estimation of missing data.