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Best-possible bounds on sets of bivariate distribution functions

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Listed:
  • Nelsen, Roger B.
  • Molina, José Juan Quesada
  • Lallena, José Antonio Rodríguez
  • Flores, Manuel Úbeda

Abstract

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.

Suggested Citation

  • Nelsen, Roger B. & Molina, José Juan Quesada & Lallena, José Antonio Rodríguez & Flores, Manuel Úbeda, 2004. "Best-possible bounds on sets of bivariate distribution functions," Journal of Multivariate Analysis, Elsevier, vol. 90(2), pages 348-358, August.
    • Handle: RePEc:eee:jmvana:v:90:y:2004:i:2:p:348-358
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    References listed on IDEAS

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    1. Alsina, Claudi & Nelsen, Roger B. & Schweizer, Berthold, 1993. "On the characterization of a class of binary operations on distribution functions," Statistics & Probability Letters, Elsevier, vol. 17(2), pages 85-89, May.
    2. Genest, C. & Quesada Molina, J. J. & Rodriguez Lallena, J. A. & Sempi, C., 1999. "A Characterization of Quasi-copulas," Journal of Multivariate Analysis, Elsevier, vol. 69(2), pages 193-205, May.
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    Cited by:

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    2. Arnold, Sebastian & Molchanov, Ilya & Ziegel, Johanna F., 2020. "Bivariate distributions with ordered marginals," Journal of Multivariate Analysis, Elsevier, vol. 177(C).
    3. Nelsen, Roger B. & Úbeda-Flores, Manuel, 2012. "How close are pairwise and mutual independence?," Statistics & Probability Letters, Elsevier, vol. 82(10), pages 1823-1828.
    4. Kaas, Rob & Laeven, Roger J.A. & Nelsen, Roger B., 2009. "Worst VaR scenarios with given marginals and measures of association," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 146-158, April.
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    7. Yanqin Fan & Carlos A. Manzanares, 2017. "Partial identification of average treatment effects on the treated through difference-in-differences," Econometric Reviews, Taylor & Francis Journals, vol. 36(6-9), pages 1057-1080, October.
    8. Peter Tankov, 2010. "Improved Frechet bounds and model-free pricing of multi-asset options," Papers 1004.4153, arXiv.org, revised Mar 2011.
    9. Jonathan Ansari & Eva Lutkebohmert & Ariel Neufeld & Julian Sester, 2022. "Improved Robust Price Bounds for Multi-Asset Derivatives under Market-Implied Dependence Information," Papers 2204.01071, arXiv.org, revised Sep 2023.
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