On bivariate dependence and the convex order (Q1866984)

Recall that the random variables \(X\) and \(Y\) are comparable in the convex order (written as \(X\leq_{\text{cx}}Y\)) if \(Eh(X)\leq EF(Y)\) for all convex functions \(h\) for which the expectations are finite. The main result of this paper claims that if \((X_1,X_2)\) and \((Y_1,Y_2)\) are negatively dependent in the sense that each has a reverse regular of order~2 (RR\(_2\)) joint density, and if \(X_1\leq_{\text{cx}}Y_1\), \(X_2\leq_{\text{cx}}Y_2\), and \(\text{Cov}(X_1,X_2)>\text{Cov}(Y_1,Y_2)\), then \(E\max\{X_1,X_2\}\leq E\max\{Y_1,Y_2\}\).