Best-possible bounds on sets of bivariate distribution functions
The fundamental best-possible bounds inequality for bivariate distribution functions with given margins is the Frechet-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)[less-than-or-equals, slant]H(x,y)[less-than-or-equals, slant]min(F(x),G(y)) for all x,y in [-[infinity],[infinity]]. 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.
Year of publication: |
2004
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Authors: | Nelsen, Roger B. ; Molina, José Juan Quesada ; Lallena, José Antonio Rodríguez ; Flores, Manuel Úbeda |
Published in: |
Journal of Multivariate Analysis. - Elsevier, ISSN 0047-259X. - Vol. 90.2004, 2, p. 348-358
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Publisher: |
Elsevier |
Keywords: | Bounds Copulas Distribution functions Kendall's tau Quartiles Quasi-copulas |
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