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Reliability and expectation bounds for coherent systems with exchangeable components. (English) Zbl 1102.62111
Summary: Sharp upper and lower bounds are obtained for the reliability functions and the expectations of lifetimes of coherent systems based on dependent exchangeable absolutely continuous components with a given marginal distribution function, by use of the concept of F. J. Samaniego’s signature [IEEE Trans. Reliab. 34, 69–72 (1985; Zbl 0585.62169)]. We first show that the distribution of any coherent system based on exchangeable components with absolutely continuous joint distribution is a convex combination of distributions of order statistics (equivalent to the $$k$$-out-of-$$n$$ systems) with the weights identical with the values of the Samaniego signature of the system. This extends the Samaniego representation valid for the case of independent and identically distributed components. Combining the representation with optimal bounds on linear combinations of distribution functions of order statistics from dependent, identically distributed samples, we derive the corresponding reliability and expectation bounds, dependent on the signature of the system and marginal distribution of dependent components.
We also present the sequences of exchangeable absolutely continuous joint distributions of components which attain the bounds in the limit. As an application, we obtain the reliability bounds for all the coherent systems with three and four exchangeable components, expressed in terms of the parent marginal reliability function and specify the respective expectation bounds for exchangeable exponential components, comparing them with the lifetime expectations of systems with independent and identically distributed exponential components.

##### MSC:
 62N05 Reliability and life testing 62E15 Exact distribution theory in statistics 60E15 Inequalities; stochastic orderings 62G30 Order statistics; empirical distribution functions
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