Modeling directional monotonicity with copulas
Enrique de Amo, David García-Fernández, José Juan Quesada-Molina, Manuel Úbeda-Flores
TL;DR
This work introduces monotonicity according to a direction $I(alpha)$ as a directional dependence notion within copula theory, then derives explicit inequalities that characterize $I(alpha)$ for the bivariate, trivariate, and general multivariate cases. The authors prove a central multivariate result linking the $n$-copula $C$ and its marginal marginals via an alternating-sum inequality that must hold for all $u\le u'$, thereby providing a testable, dimension-free framework. The paper also shows several concrete examples, including $M^n$, $Pi^n$, and generalized FGM families, illustrating when different directional patterns hold. Overall, the approach yields a rigorous, copula-based description of directional monotonicity with potential applications to dependence modeling and distribution construction in high dimensions.
Abstract
The purpose of this paper is to characterize the concept of monotonicity according to a direction related to a set of n random variables in terms of its associated n-copula C. We start establishing relationships in the bivariate and trivariate cases, which help to understand the extension to the multivariate case. Examples of copulas in all the studied cases are provided.