Best-possible bounds on sets of bivariate distribution functions

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(...

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Bibliographic Details
Published in:Journal of multivariate analysis 2004-08, Vol.90 (2), p.348-358
Main Authors: Nelsen, Roger B, Molina, José Juan Quesada, Lallena, José Antonio Rodrı́guez, Flores, Manuel Úbeda
Format: Article
Language:English
Subjects:
Bounds
Bounds Copulas Distribution functions Kendall's tau Quartiles Quasi-copulas
Copulas
Distribution functions
Exact sciences and technology
Kendall's tau
Mathematics
Multivariate analysis
Probability and statistics
Quartiles
Quasi-copulas
Random variables
Sciences and techniques of general use
Statistics
Studies
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Summary: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.
ISSN:0047-259X
1095-7243
DOI:10.1016/j.jmva.2003.09.002