The fundamental best-possible bounds inequality for bivariate distribution functions with given margins is the Fre´chet-Hoeffding inequality: If H denotes the joint distribution function of random variables X and Y whose margins are F and G; respectively, then maxð0; F ðxÞ þ GðyÞ À 1ÞpHðx; yÞpminðF ðxÞ; GðyÞÞ for all x; y in ½ÀN; N: In this paper we employ copulas and quasi-copulas to find similar best-possible bounds on arbitrary sets of bivariate distribution functions with given margins. As an application, we discuss bounds for a bivariate distribution function H with given margins F and G when the values of H are known at quartiles of X and Y : r
Using an iterated function system, the authors construct families of copulas whose supports are fractals. In particular, they show that the members of one family have supports with arbitrary Hausdorff dimensions in the interval (1,2). They also employ those copulas to construct more general bivariate distribution functions with fractal supports.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.