Bounds on expectation of order statistics from a finite population

Abstract

Consider a simple random sample X1,X2,...,Xn, taken without replacement from a finite ordered population ∏ = {x1 ≤ x2 ≤ ⋯ ≤ XN} (n ≤ N), where each element of ∏ has equal probability to be chosen in the sample. Let X1:n ≤ X2:n ≤ ⋯ ≤ Xn:n be the ordered sample. In the present paper, the best possible bounds for the expectations of the order statistics Xi:n (1 ≤ i ≤ n) and the sample range Rn = Xn:n - X1:n are derived in terms of the population mean and variance. Some results are also given for the covariance in the simplest case where n = 2. An interesting feature of the bounds derived here is that they reduce to some well-known classical results (for the i.i.d. case) as N → ∞. Thus, the bounds established in this paper provide an insight into Hartley-David-Gumbel, Samuelson-Scott, Arnold-Groeneveld and some other bounds. © 2002 Elsevier Science B.V. All rights reserved.