In this article, we study the problem of finding tight bounds on the
expected value of the kth-order statistic E
[Xk:n] under first and
second moment information on n real-valued random variables.
Given means E [Xi] =
μi and variances
σi2, we show that the tight upper bound
on the expected value of the highest-order statistic E
[Xn:n] can be computed
with a bisection search algorithm. An extremal discrete distribution is
identified that attains the bound, and two closed-form bounds are
proposed. Under additional covariance information
= Qij, we show that the tight upper bound on
the expected value of the highest-order statistic can be computed with
semidefinite optimization. We generalize these results to find bounds on
the expected value of the kth-order statistic under mean and
variance information. For k < n, this bound is shown
to be tight under identical means and variances. All of our results are
distribution-free with no explicit assumption of independence made.
Particularly, using optimization methods, we develop tractable approaches
to compute bounds on the expected value of order statistics.