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On distributions of order statistics for absolutely continuous copulas with applications to reliability. (English) Zbl 1180.60013
Two main results are given concerning the marginal distribution functions of order statistics that are based on $$n$$ dependent standard uniform random variables having a joint absolutely continuous distribution (i.e. whose joint distribution function is an absolutely continuous copula). Specifically, Theorem 2 gives necessary and sufficient conditions that ensure that a vector of distribution functions $$(G_1,G_2,\dots,G_n)$$ corresponds to the vector of distribution functions of consecutive order statistics. Theorem 4 characterizes a univariate distribution function $$G$$ as the distribution function of a given order statistics. The presented problems are connected with the study of coherent systems in reliability theory.

##### MSC:
 60E05 Probability distributions: general theory 62G30 Order statistics; empirical distribution functions 62H05 Characterization and structure theory for multivariate probability distributions; copulas 62N05 Reliability and life testing
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