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Bivariate distributions with ordered marginals

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  • Arnold, Sebastian
  • Molchanov, Ilya
  • Ziegel, Johanna F.

Abstract

This paper provides a characterization of all possible dependency structures between two stochastically ordered random variables. The answer is given in terms of copulas that are compatible with the stochastic order and the marginal distributions. The extremal values for Kendall’s τ and Spearman’s ρ for all these copulas are given in closed form. We also find an explicit form for the joint distribution with the maximal entropy. A multivariate extension and a generalization to random elements in partially ordered spaces are also provided.

Suggested Citation

  • Arnold, Sebastian & Molchanov, Ilya & Ziegel, Johanna F., 2020. "Bivariate distributions with ordered marginals," Journal of Multivariate Analysis, Elsevier, vol. 177(C).
    • Handle: RePEc:eee:jmvana:v:177:y:2020:i:c:s0047259x19300831
    DOI: 10.1016/j.jmva.2019.104585
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    References listed on IDEAS

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    1. Paul Embrechts & Marius Hofert, 2013. "A note on generalized inverses," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(3), pages 423-432, June.
    2. 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.
    3. Navarro, Jorge & Spizzichino, Fabio, 2010. "On the relationships between copulas of order statistics and marginal distributions," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 473-479, March.
    4. Dietz, Markus & Fuchs, Sebastian & Schmidt, Klaus D., 2016. "On order statistics and their copulas," Statistics & Probability Letters, Elsevier, vol. 117(C), pages 165-172.
    5. Lebrun, Régis & Dutfoy, Anne, 2014. "Copulas for order statistics with prescribed margins," Journal of Multivariate Analysis, Elsevier, vol. 128(C), pages 120-133.
    6. Butucea, Cristina & Delmas, Jean-François & Dutfoy, Anne & Fischer, Richard, 2015. "Maximum entropy copula with given diagonal section," Journal of Multivariate Analysis, Elsevier, vol. 137(C), pages 61-81.
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    Cited by:

    1. Chen, Yuyu & Lin, Liyuan & Wang, Ruodu, 2022. "Risk aggregation under dependence uncertainty and an order constraint," Insurance: Mathematics and Economics, Elsevier, vol. 102(C), pages 169-187.
    2. Yuyu Chen & Liyuan Lin & Ruodu Wang, 2021. "Risk Aggregation under Dependence Uncertainty and an Order Constraint," Papers 2104.07718, arXiv.org, revised Oct 2021.

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