A broader look at bivariate distributions applicable in hydrology

In hydrology, one often needs to model two dependent variables; regardless of the marginal distribution that each of them follows, while still taking account of the dependence structure that exists between them. In this respect, the use of copulas to model the dependence between the two variables has been found to be attractive. However, the use of copulas for bivariate modeling in previous applications has been restrictive, in the sense that it has focused exclusively on copulas expressible by a simple mathematical form, while ignoring a wide class of other bivariate models that are available to the practitioner. These other models were not given the attention that they deserve, because it was wrongly assumed that they are not flexible enough to allow for different marginal distribution families to be chosen for the two variables that are being modeled. In this article, we shed more light on this class of models, by presenting three specific members of this class, and showing their usefulness through hydrological applications. The results of these applications show that the suggested models have their own dependence structures, and need to be given the same level of consideration in hydrology as the more popular copula models. There is in fact a need in hydrology to consider a wide variety of bivariate models, especially if each model has its own distinctive dependence structure, and given the wide variety of dependence structures that the hydrologist might encounter. The main purpose of this paper is therefore to encourage the exploration of a wider class of potentially useful bivariate models that hydrologists need to consider.