Quantile curves and dependence structure for bivariate distributions

Within the context of a general bivariate distribution an intuitive method is presented in order to study the dependence structure of the two distributions. A set of points—level curve—which accumulate the same probability for a fixed quadrant is considered. This procedure provides four level curves which can be considered as the boundary of a generalization of the real interquantile interval. It is shown that the accumulated probability among the level curves depends on the dependence structure of the distribution function where the dependence structure is given by the notion of copula. Furthermore, the case when the marginal distributions are independent is investigated. This result is used to find out positive or negative dependence properties for the variables. Finally, a nonparametric test for independence with a local dependence meaning is performed and applied to different data sets.