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

Abstract

Let $\mathbb{X}=(X_1,\dots ,X_n)$ be an $n$-tuple of random variables where each $X_j$ has the same known distribution function $F$ and where there is a number $k\le n$ such that for each $i\in \{1,\dots ,k\},$ all $i$-tuples have copulas with the same known diagonal ${\delta }_i$. A reliability system with such nonnegative component lifetimes $X_1,\dots ,X_n$ is a system with the property that for each $i\le k$, 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 $X_j$’s, and apply it to derive two-sided bounds (depending on $F$ and ${\delta }_i$’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 ${\delta }_2$’s, we determine stochastically extremal distributions of single order statistics.