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TIGHT BOUNDS ON EXPECTED ORDER STATISTICS

Published online by Cambridge University Press:  19 September 2006

Dimitris Bertsimas ,
Karthik Natarajan  and
Chung-Piaw Teo
Dimitris Bertsimas
Affiliation:
Sloan School of Management and Operations Research Center, Massachusetts Institute of Technology, Cambridge, MA 02139, E-mail: dbertsim@mit.edu
Karthik Natarajan
Affiliation:
Department of Mathematics, National University of Singapore, Singapore 117543, E-mail: matkbn@nus.edu.sg
Chung-Piaw Teo
Affiliation:
Department of Decision Sciences, NUS Business School, Singapore 117591, E-mail: bizteocp@nus.edu.sg
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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] = 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 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.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 20 , Issue 4 , October 2006 , pp. 667 - 686
Copyright
© 2006 Cambridge University Press

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