TY - JOUR

T1 - Limit distributions of the upper order statistics for the Lévy-frailty Marshall-Olkin distribution

AU - Barrera, Javiera

AU - Lagos, Guido

N1 - Funding Information:
Javiera Barrera acknowledges the financial support of FONDECYT grant 1161064 and of Programa Iniciativa Científica Milenio NC120062. Guido Lagos acknowledges the financial support of FONDECYT grant 3180767, Programa Iniciativa Científica Milenio NC120062, and the Center for Mathematical Modeling of Universidad de Chile through their grants Proyecto Basal PFB-03 and PIA Fellowship AFB170001.
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

VL - 23

SP - 603

EP - 628

JO - Extremes

JF - Extremes

SN - 1386-1999

IS - 4

ER -