Extremal properties of order statistic distributions for dependent samples with partially known multidimensional marginals

Let X=(X1,…,Xn) be an n-tuple of random variables where each Xj has the same known distribution function F and where there is a number k≤n such that for each i∈{1,…,k}, all i-tuples have copulas with the same known diagonal δi. A reliability system with such nonnegative component lifetimes X1,…,Xn is a system with the property that for each i≤k, all of its structurally identical sub-systems of i components have the same known reliability function. We provide a characterization for empirical distributions from the Xj’s, and apply it to derive two-sided bounds (depending on F and δi’s) for arbitrary linear combinations of distribution functions of the associated order statistics as well as to establish necessary and sufficient conditions for uniform sharpness of these bounds. Moreover, for k=2 and some classes of δ2’s, we determine stochastically extremal distributions of single order statistics.