The Marshall-Olkin (MO) distribution is considered a key model in reliability theory and in risk analysis, where it is used to model the lifetimes of dependent components or entities of a system and dependency is induced by “shocks” that hit one or more components at a time. Of particular interest is the Lévy-frailty subfamily of the Marshall-Olkin (LFMO) distribution, since it has few parameters and because the nontrivial dependency structure is driven by an underlying Lévy subordinator process. The main contribution of this work is that we derive the precise asymptotic behavior of the upper order statistics of the LFMO distribution. More specifically, we consider a sequence of n univariate random variables jointly distributed as a multivariate LFMO distribution and analyze the order statistics of the sequence as n grows. Our main result states that if the underlying Lévy subordinator is in the normal domain of attraction of a stable distribution with index of stability α then, after certain logarithmic centering and scaling, the upper order statistics converge in distribution to a stable distribution if α > 1 or a simple transformation of it if α ≤ 1. Our result can also give easily computable confidence intervals for the last failure times, provided that a proper convergence analysis is carried out first.

Marshall-Olkin（MO）分布在可靠性理论和风险分析中被认为是关键模型，在该模型中，Marshall-Olkin分布用于对系统的依赖组件或实体的寿命进行建模，而依赖是由撞击一个或多个的“冲击”引起的一次组件。特别令人感兴趣的是Marshall-Olkin（LFMO）分布的Lévy脆弱子族，因为它的参数很少，并且非平凡的依存结构是由底层Lévy隶属者过程驱动的。这项工作的主要贡献在于，我们得出了LFMO分布的高阶统计量的精确渐近行为。更具体地说，我们考虑n的序列单变量随机变量联合作为多变量LFMO分布进行分布，并随着n的增长分析序列的顺序统计量。我们的主要结果表明，如果下层的Lévy隶属子处于具有稳定指数α的稳定分布的正态吸引域中，则在经过一定对数定心和定标后，如果α > 1 ，则高阶统计收敛于稳定分布或者它的简单改造，如果α ≤1，我们的结果也能轻易放弃最后失败次数可计算置信区间，提供了一个适当的收敛性分析首先进行。