TY - JOUR
T1 - Limit distributions of the upper order statistics for the Lévy-frailty Marshall-Olkin distribution
AU - Barrera, Javiera
AU - Lagos, Guido
N1 - Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2020/12/1
Y1 - 2020/12/1
N2 - 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.
AB - 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.
KW - Dependent random variables
KW - Extreme-value theory
KW - Marshall-Olkin distribution
KW - Reliability
KW - Upper order statistics
UR - http://www.scopus.com/inward/record.url?scp=85089152676&partnerID=8YFLogxK
U2 - 10.1007/s10687-020-00386-z
DO - 10.1007/s10687-020-00386-z
M3 - Article
AN - SCOPUS:85089152676
SN - 1386-1999
VL - 23
SP - 603
EP - 628
JO - Extremes
JF - Extremes
IS - 4
ER -