Stochastic comparisons of lifetimes of fail-safe systems with dependent and heterogeneous components under random shocks

A fail-safe system is a (n - 1)-out-of-n system whose lifetime is represented by the second-order statistic. This work studies stochastic comparisons of lifetimes of fail-safe systems with dependent and heterogeneous components, where the components are subjected to random shocks instantaneously. The results are derived for a general semiparametric family of distributions of the component lifetimes. We derive sufficient conditions for the comparison results under the usual stochastic order, and we provide counterexamples to show that under those conditions, similar comparisons cannot be made under the hazard rate order.