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

TL;DR

This paper derives the tightest possible upper bound for the expected range of dependent random variables with specified means and variances, characterizing extremal distributions and comparing with existing bounds.

Contribution

It provides an explicit characterization of extremal distributions that attain the maximum expected range under mean-variance constraints.

Findings

Derived the best possible upper bound for the expected range.

Characterized extremal distributions that achieve the bound.

Compared the new bound with existing results.

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.