On stochastic ordering among extreme shock models

In the usual shock models, the shocks arrive from a single source. Bozbulut and Eryilmaz [(2020). Generalized extreme shock models and their applications. Communications in Statistics – Simulation and Computation49(1): 110–120] introduced two types of extreme shock models when the shocks arrive from one of $m\geq 1$ possible sources. In Model 1, the shocks arrive from different sources over time. In Model 2, initially, the shocks randomly come from one of $m$ sources, and shocks continue to arrive from the same source. In this paper, we prove that the lifetime of Model 1 is less than Model 2 in the usual stochastic ordering. We further show that if the inter-arrival times of shocks have increasing failure rate distributions, then the usual stochastic ordering can be generalized to the hazard rate ordering. We study the stochastic behavior of the lifetime of Model 2 with respect to the severity of shocks using the notion of majorization. We apply the new stochastic ordering results to show that the age replacement policy under Model 1 is more costly than Model 2.