New properties of the orthant convex-type stochastic orders

The orthant convex and concave orders have been studied in the literature as extensions of univariate variability orders. In this paper, new results are proposed for bivariate orthant convex-type orders between vectors. In particular, we prove that these orders cannot be considered as dependence orders since they fail to verify several desirable properties that any positive dependence order should satisfy. Among other results, the relationships between these orders under certain transformations are presented, as well as that the orthant convex orders between bivariate random vectors with the same means are sufficient conditions to order the corresponding covariances. We also show that establishing the upper orthant convex or lower orthant concave orders between two vectors in the same Fréchet class is not equivalent to establishing these orders between the corresponding copulas except when marginals are uniform distributions. Several examples related with concordance measures, such as Kendall’s tau and Spearman’s rho, are also given, as are results on mixture models.