Comonotonicity and counter-monotonicity: Review and implications for likelihood-based estimation

Comonotonicity and counter-monotonicity refer to the strongest possible form of dependence, namely perfect positive and negative dependence, respectively. For continuous random vectors, comonotonicity implies a functional relation between the components and, hence a reduction in the dimensionality of the problem. The case of variables with discrete margins is much more complex, however. The goal of this article is to review the notion of perfect dependence, with a particular emphasis on the discrete case, as well as its implications for likelihood-based estimation.