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A general multivariate new better than used (MNBU) distribution and its properties. (English) Zbl 1466.60025

The authors consider a multivariate shock model for a system made up of \(n\) components. Shocks arrive at the points of a generalised Pólya process, and can either affect a single system component or all \(n\) components simultaneously. Components affected by a shock immediately fail. The random variables of interest here are the time \(X_i\) (\(i=1,\ldots,n\)) until the failure of the \(i\)th component, and \(X=\min_iX_i\). Based on an explicit expressions they derive for the joint survival function of the \(X_i\), the authors investigate positive dependence properties of these random variables. For example, the \(X_i\) are shown to satisfy the positive upper orthant dependency property, and under certain conditions (that the underlying intensity function of the shocks is increasing in time and that the probability a shock affects a given component is constant in time) the \(X_i\) are multivariate new better than used. The authors conclude the paper with a discussion of the consequences of these results for the time \(X\) until failure of the system.

MSC:

60E05 Probability distributions: general theory
60E15 Inequalities; stochastic orderings
62H05 Characterization and structure theory for multivariate probability distributions; copulas
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