Maximizing the expected range from dependent observations under mean–variance information

Abstract

In this article we derive the best possible upper bound for E[maxi{Xi} − mini{Xi}] under given means and variances on n random variables Xi. The random vector (X1, . . . , Xn) is allowed to have any dependence structure, provided EXi = μi and Var Xi = σi 2 , 0 < σi < ∞ We provide an explicit characterization of the n-variate distributions that attain the equality (extremal random vectors), and the tight bound is compared to other existing results. © 2015 Taylor & Francis.