Metrika (2003) 58: 159–171 DOI 10.1007/s001840200234 > Springer-Verlag 2003 Sharp exponential and entropy bounds on expectations of generalized order statistics M. Kaluszka and A. Okolewski Institute of Mathematics, Technical University of Lodz, ul. Zwirki 36, 90-924 Lodz, Poland Received October 2001/Revised May 2002 Abstract. Sharp lower and upper bounds on expected values of generalized order statistics are proven by the use of Moriguti’s inequality combined with the Young inequality. The bounds are expressed in terms of exponential moments or entropy. They are attainable providing new characterizations of some nontrivial distributions. AMS subject classification: 62G30, 62H10 Key words and phrases: generalized order statistics; order statistics; records; entropy; Moriguti’s inequality; Young’s inequality; Gumbel distribution; Pareto distribution 1. Introduction Let X ; X 1 ; X 2 ; ... be i.i.d. random variables with a common distribution func- tion F. Let X r:n denote the rth order statistic (OS, for short) from the sample X 1 ; ... ; X n . The kth record statistics (RS’s, for short) Y ðkÞ r are defined by Y ðkÞ r ¼ X L k ðrÞ: L k ðrÞþk1 ; r ¼ 1; 2; ... ; k ¼ 1; 2; ... ; where L k ð1Þ¼ 1, L k ðr þ 1Þ¼ minf j : X L k ðrÞ: L k ðrÞþk1 < X j :j þk1 g for r ¼ 1; 2; ... (cf. Dziubdziela and Kopocin ´ ski, 1976). Define the quantile function F 1 ðtÞ¼ inf fs A R : F ðsÞ b tg, t A ð0; 1Þ. The generalized order statistics are defined by Kamps (1995) as follows: Definition 1. Let r; n A N, k > 0, m 2 R be parameters such that h r ¼ k þ ðn  rÞðm þ 1Þ b 1 for all r A f1; ... ; ng. If the random variables U ðr; n; m; kÞ, r ¼ 1; ... ; n, possess a joint density function of the form