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Barrera, J., Lagos, G. (2020). Limit distributions of the upper order statistics for the Levyfrailty MarshallOlkin distribution. Extremes, 23, 603–628.
Abstract: The MarshallOlkin (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 Levyfrailty subfamily of the MarshallOlkin (LFMO) distribution, since it has few parameters and because the nontrivial dependency structure is driven by an underlying Levy 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 ofnunivariate random variables jointly distributed as a multivariate LFMO distribution and analyze the order statistics of the sequence asngrows. Our main result states that if the underlying Levy 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 <= 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.

Matus, O., Barrera, J., Moreno, E., & Rubino, G. (2019). On the MarshallOlkin Copula Model for Network Reliability Under Dependent Failures. IEEE Trans. Reliab., 68(2), 451–461.
Abstract: The MarshallOlkin (MO) copulamodel has emerged as the standard tool for capturing dependence between components in failure analysis in reliability. In this model, shocks arise at exponential random times, that affect one or several components inducing a natural correlation in the failure process. However, because the number of parameter of the model grows exponentially with the number of components, MO suffers of the “curse of dimensionality.” MO models are usually intended to be applied to design a network before its construction; therefore, it is natural to assume that only partial information about failure behavior can be gathered, mostly from similar existing networks. To construct such an MO model, we propose an optimization approach to define the shock's parameters in the MO copula, in order to match marginal failures probabilities and correlations between these failures. To deal with the exponential number of parameters of this problem, we use a columngeneration technique. We also discuss additional criteria that can be incorporated to obtain a suitable model. Our computational experiments show that the resulting MO model produces a close estimation of the network reliability, especially when the correlation between component failures is significant.
