Xiaohu Li

1. As one building block of the coherent system, the k-out-of-n structure, which operates whenever at least k components function properly, has been paid much attention in the literature of enginee...

The stochastic ordering of order statistics plays a central role in many fields involving probability and statistics. Over the past decade, significant progress has been made in ordering the first and second order statistics from heterogeneous or statistically dependent observations. This paper further develops necessary and sufficient conditions for the third-largest and third-smallest order statistics from observations linked by an Archimedean copula and following, respectively, the proportional hazards, proportional reverse hazards, and scale models. The characterization results and sufficient conditions presented herein not only enrich the existing literature but also provide novel theoretical insights for applications in fields such as reliability systems, multiple outlier models, and the design of third-price auction mechanisms.

System components usually attain marginal lifetimes with stochastic dependence in the context of load-sharing reliability structures. This study deals with the load-sharing parallel systems of two components. We prove that two marginal lifetimes are positively quadrant dependent when component lifetimes have continuous probability distributions, and such a stochastic dependence is upgraded to the total positive of order 2 in the setting of component lifetimes having an exponential distribution. In addition, we discuss how these findings shed light on related results for the load-sharing Ross model, the conditional residual lifetime, and the conditional inactivity time.

For coherent systems with components and active redundancies having heterogeneous and dependent lifetimes, we prove that the lifetime of system with redundancy at component level is stochastically larger than that with redundancy at system level. In particular, in the setting of homogeneous components and redundancy lifetimes linked by an Archimedean survival copula, we develop sufficient conditions for the reversed hazard rate order, the hazard rate order and the likelihood ratio order between two system lifetimes, respectively. The present results substantially generalize some related results in the literature. Several numerical examples are presented to illustrate the findings as well.