Bounds on expectation of order statistics from a finite population

• Published in: Journal of statistical planning and inference 2003-05, Vol.113 (2), p.569-588
• Main Authors: Balakrishnan, N., Charalambides, C., Papadatos, N.
• Format: Article
• Language: English
• Subjects: Distribution theory; Exact sciences and technology; Expectation bounds; Finite population; Mathematics; Nonparametric inference; Order statistics; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Statistics; Without replacement scheme
• ISSN: 0378-3758; 1873-1171
• DOI: 10.1016/S0378-3758(01)00321-4

Consider a simple random sample X 1, X 2,…, X n , taken without replacement from a finite ordered population Π={ x 1⩽ x 2⩽⋯⩽ x N } ( n⩽ N), where each element of Π has equal probability to be chosen in the sample. Let X 1: n ⩽ X 2: n ⩽⋯⩽ X n: n be the ordered sample. In the present paper, the best possible bounds for the expectations of the order statistics X i :n (1⩽i⩽n) and the sample range R n = X n: n − X 1: 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.