Bounds on expectation of order statistics from a finite population

Consider a simple random sample X1,X2,…,Xn, taken without replacement from a finite ordered population Π={x1⩽x2⩽⋯⩽xN} (n⩽N), where each element of Π 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.