Limit distributions of the upper order statistics for the L\'evy-frailty Marshall-Olkin distribution

Abstract

The Marshall-Olkin (MO) distribution has been 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 L\'evy-frailty subfamily of the Marshall-Olkin (LFMO) distribution, since it has few parameters and because the nontrivial dependency structure is driven by an underlying L\'evy subordinator process. The main contribution of our 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 L\'evy subordinator is in the normal domain of attraction of a stable distribution with index of stability $\alpha$ then, after certain logarithmic centering and scaling, the upper order statistics converge in distribution to a stable distribution if $\alpha>1$ or a simple transformation of it if $\alpha\leq1$. Our result is especially useful in network reliability and systemic risk, when modeling the lifetimes of components in a system using the LFMO distribution, as it allows to understand the behavior of systems that rely on its last working components. Our result can also give easily computable confidence intervals for these components, provided that a proper convergence analysis is carried out first.