  Motivated by recently investigated results on dependence measures and robust
risk models, this paper provides an overview of dependence properties of many
well-known bivariate copula families, where the focus is on the Schur order for
conditional distributions, which has the fundamental property that minimal
elements characterize independence and maximal elements characterize perfect
directed dependence. We give conditions on copulas that imply the Schur
ordering of the associated conditional distribution functions. For
extreme-value copulas, we prove the equivalence of the lower orthant order, the
Schur order for conditional distributions, and the pointwise order of the
associated Pickands dependence functions. Further, we provide several tables
and figures that list and illustrate various positive dependence and
monotonicity properties of copula families, in particular, from classes of
Archimedean, extreme-value, and elliptical copulas. Finally, for Chatterjee's
rank correlation, which is consistent with the Schur order for conditional
distributions, we give some new closed-form formulas in terms of the parameter
of the underlying copula family.
