On the interplay between variability and negative dependence for bivariate distributions. (Q1413897)

The author identifies a mistake in the main result in the paper by \textit{A. Paul} [ibid. 30, 181--188 (2002; Zbl 1027.60012)]. He proves, however, that if \({X}=(X_1,X_2)\) and \({Y}=(Y_1,Y_2)\) have a common conditionally decreasing copula, and if \(X_1\leq_{\text{cx}}Y_1\) and \(X_2\leq_{\text{cx}}Y_2\) (where \(\leq_{\text{cx}}\) denotes the convex stochastic order), then \(Ef(X_1,X_2)\leq Ef(Y_1,Y_2)\) for all submodular and componentwise convex functions \(f\). Some related results are also included.