A Survey of Dynamic Representations and Generalizations of the Marshall–Olkin Distribution

Bernhart, German; Fernández, Lexuri; Mai, Jan-Frederik; Schenk, Steffen; Scherer, Matthias

doi:10.1007/978-3-319-19039-6_1

Cited by 6 publications

(4 citation statements)

References 29 publications

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“…Nonetheless, an important shortfall of the general MO model is that it requires to specify a number of parameters that is exponential in the number of components when no further assumptions or simplifications are made. In some real-world situations, the nature of the problem can lead to natural simplifications that reduce the number of parameters, e.g., in portfolio-credit risks or in insurance; see [3]. In this line, [24] propose a Lasso selection model to obtain the parameters.…”

Section: Literature Reviewmentioning

confidence: 99%

“…. , n}, where Ψ(x) = − log Ee −xS1 is the Laplace exponent of the Lévy subordinator S; see, e.g., [3]. Note that the latter parameters λ V only depend on |V |, thus in particular making the components exchangeable; see [19].…”

Section: Marshall-olkin and Lévy-frailty Marshall-olkin Distributionsmentioning

confidence: 99%

“…We also remark that the MO distribution in n dimensions of Definition 1 requires specifying a number of 2 n − 1 parameters λ V , whereas the LFMO distribution only requires parameterizing the Lévy subordinator S. In this sense, the LFMO distribution is a way to reduce the parametric complexity of the MO distribution by means of introducing the latent variable S. In fact, most characteristics of the LFMO model are expressed in terms of S 1 and its Laplace exponent Ψ(x) = − log Ee −xS1 , which have, respectively, rich probabilistic and analytical structures that allow to study and exploit the model in great detail; see, e.g., [3,12]. Finally, it is worth mentioning that [8] studies the LFMO construction when the triggers are nonhomogeneous, that is, when the triggers in Definition 2 are independent and exponential but with different rates.…”

Section: Marshall-olkin and Lévy-frailty Marshall-olkin Distributionsmentioning

confidence: 99%

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

Barrera,

Lagos

2018

Preprint

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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 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 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 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 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.

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Section: Literature Reviewmentioning

confidence: 99%

Section: Marshall-olkin and Lévy-frailty Marshall-olkin Distributionsmentioning

confidence: 99%

Section: Marshall-olkin and Lévy-frailty Marshall-olkin Distributionsmentioning

confidence: 99%

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

Barrera,

Lagos

2018

Preprint

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“…In particular, the hazard rate order of a parallel system with two components having MOE has been studied in [6]. The MOE distribution has been generalized in different ways due to its practical use; one may refer to the monograph by Bernhart [11].…”

Section: Introductionmentioning

confidence: 99%

Stochastic Comparisons of Largest-Order Statistics and Ranges from Marshall–Olkin Bivariate Exponential and Independent Exponential Variables

Balakrishnan,

Saadat Kia (Barmalzan),

Mehrpooya

2023

Symmetry

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Sample range and the associated functions such as survival function and mean residual life function have found many important applications in the reliability field. In this work, we establish some results that are in two different directions. In the first part, we establish some conditions for comparing the largest-order statistics (in the sense of mean residual life order) arising from bivariate Marshall–Olkin exponential distribution. Then, in the second part, we present some sufficient conditions for comparing sample ranges (in the sense of usual stochastic order and reversed hazard rate order) arising from independent exponential random variables.

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Limiting Behavior of Mixed Coherent Systems With Lévy‐Frailty Marshall–Olkin Failure Times

Lagos,

Barrera,

Romero

et al. 2024

Appl Stoch Models Bus & Ind

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In this article, we show a limit result for the reliability function of a system—that is, the probability that the whole system is still operational after a certain given time—when the number of components of the system grows to infinity. More specifically, we consider a sequence of mixed coherent systems whose components are homogeneous and non‐repairable, with failure‐times governed by a Lévy‐Frailty Marshall–Olkin (LFMO) distribution—a distribution that allows simultaneous component failures. We show that under integrability conditions the reliability function converges to the probability of a first‐passage time of a Lévy subordinator process. To the best of our knowledge, this is the first result to tackle the asymptotic behavior of the reliability function as the number of components of the system grows. To illustrate our approach, we give an example of a parametric family of reliability functions where the system failure time converges in distribution to an exponential random variable, and give computational experiments testing convergence.

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A Survey of Dynamic Representations and Generalizations of the Marshall–Olkin Distribution

Cited by 6 publications

References 29 publications

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

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

Stochastic Comparisons of Largest-Order Statistics and Ranges from Marshall–Olkin Bivariate Exponential and Independent Exponential Variables

Limiting Behavior of Mixed Coherent Systems With Lévy‐Frailty Marshall–Olkin Failure Times

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