| dc.description.abstract | 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 compo_x005F_x0002_nents 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 Levy-frailty subfamily of ´ the Marshall-Olkin (LFMO) distribution, since it has few parameters and because the nontrivial dependency structure is driven by an underlying Levy subordinator pro- ´ cess. 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 mul_x005F_x0002_tivariate LFMO distribution and analyze the order statistics of the sequence as n grows. Our main result states that if the underlying Levy subordinator is in the nor- ´ mal 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 distri_x005F_x0002_bution 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 | |
