Tight Bounds on Expected Order Statistics

In this paper, we study the problem of finding tight bounds on the expected value of the kth order statistic E[xk:n] under moment information on n real-valued random variables. Given means E[xi] = μi and variances V ar[xi] = σ i , 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 new closed form bounds are proposed. Under additional covariance information Cov[xi, xj ] = 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 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.