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

Abstract

In this article we derive the best possible upper bound for $E[\max X_i-{\min }_i{X}_i]$ under given means and variances on $n$ random variables $X_i$. The random vector $(X_1,...,X_n)$ is allowed to have any dependence structure, provided $E{X}_i={\mu }_i$ and $Var{X}_i={\sigma }_i^2$, $0<{\sigma }_i<\infty$. 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. Key words and phrases: Range; Dependent Observations; Tight Expectation Bounds; Extremal Random Vectors; Probability Matrices; Characterizations.

arXiv comment34 pages