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|Title:||Tight bounds on expected order statistics|
|Citation:||Bertsimas, D.,Natarajan, K.,Teo, C.-P. (2006). Tight bounds on expected order statistics. Probability in the Engineering and Informational Sciences 20 (4) : 667-686. ScholarBank@NUS Repository.|
|Abstract:||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 Var [Xi] = σi 2, 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 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 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. © 2006 Cambridge University Press.|
|Source Title:||Probability in the Engineering and Informational Sciences|
|Appears in Collections:||Staff Publications|
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