Bounds on expectations of order statistics via extremal dependences

Using the concept of r-extremal dependence, which generalizes Lai and Robbins (1976) maximal dependence, we propose alternative proofs and some new results concerning expectations of order statistics with any rank, from possibly dependent variates. In particular, new, distribution-free and tight bounds are given for the expectations of order statistics from i.d. variates whose common distribution is symmetrical. Sharp approximations of the tight bounds are also given for the standard normal distribution.