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 sev-
eral 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.

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• New properties of the orthant convex-type stochastic orders