Best-possible bounds on sets of bivariate distribution functions

Authors:
Roger B. Nelsen;José Juan Quesada Molina;José Antonio Rodríguez Lallena;Manuel Úbeda Flores
Affiliations:
Department of Mathematical Sciences, Lewis & Clark College, 0615 SW Palatine Hill Rd., Portland, OR;Deartamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain;Deartamento de Estadistica y Matemática Aplicada, Universidad de Almeria, 04120 Almería, Spain;Deartamento de Estadistica y Matemática Aplicada, Universidad de Almeria, 04120 Almería, Spain
Venue:
Journal of Multivariate Analysis
Year:
2004

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Abstract

The fundamental best-possible bounds inequality for bivariate distribution functions with given margins is the Fréchet-Hoeffding inequality: If H denotes the joint distribution function of random variables X and Y whose margins are F and G, respectively, then max(0, F(x) + G(y) - 1)≤H(x,y)≤min(F(x), G(y)) for all x,y in [-∞, ∞]. 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.