On the Maximum.Minimums Identity: Extension and Applications

For real numbers x1,.,xn the maximum.minimums identity allows us to express the maximum of x1,.,xn in terms of the minimums of subsets of {x1,.,xn}. In this note, we provide an extension allowing us to express the kth-ranked element in terms of the minimums of subsets of sizes (n.k+1),.,n. We also discuss the dual identity, allowing us to express the kth-ranked element in terms of the maximums of subsets of sizes k,.,n. We present three examples: The first relates to the expected value of order statistics from independent nonidentical geometric distributions, the second to the partial coupon collector.s problem, and the third to relations among moments of order statistics.