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Stochastic Comparison of Random Vectors with a Common Copula

Author

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  • Alfred Müller

    () (Institut für Wirtschaftstheorie und Operations Research, Universität Karlsruhe, Geb. 20.21, D-76128, Karlsruhe, Germany)

  • Marco Scarsini

    () (Dipartimento di Scienze, Università D'Annunzio, Viale Pindaro 42, I-65127 Pescara, Italy)

Abstract

We consider two random vectors X and Y , such that the components of X are dominated in the convex order by the corresponding components of Y . We want to find conditions under which this implies that any positive linear combination of the components of X is dominated in the convex order by the same positive linear combination of the components of Y . This problem has a motivation in the comparison of portfolios in terms of risk. The conditions for the above dominance will concern the dependence structure of the two random vectors X and Y , namely, the two random vectors will have a common copula and will be conditionally increasing. This new concept of dependence is strictly related to the idea of conditionally increasing in sequence, but, in addition, it is invariant under permutation. We will actually prove that, under the above conditions, X will be dominated by Y in the directionally convex order, which yields as a corollary the dominance for positive linear combinations. This result will be applied to a portfolio optimization problem.

Suggested Citation

  • Alfred Müller & Marco Scarsini, 2001. "Stochastic Comparison of Random Vectors with a Common Copula," Mathematics of Operations Research, INFORMS, vol. 26(4), pages 723-740, November.
  • Handle: RePEc:inm:ormoor:v:26:y:2001:i:4:p:723-740
    DOI: 10.1287/moor.26.4.723.10006
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    References listed on IDEAS

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    1. Moshe Shaked & J. Shanthikumar, 1990. "Parametric stochastic convexity and concavity of stochastic processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 42(3), pages 509-531, September.
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    Cited by:

    1. Marcello Basili & Paulo Casaca & Alain Chateauneuf & Maurizio Franzini, 2017. "Multidimensional Pigou–Dalton transfers and social evaluation functions," Theory and Decision, Springer, vol. 83(4), pages 573-590, December.
    2. repec:eee:ejores:v:275:y:2019:i:2:p:755-767 is not listed on IDEAS
    3. repec:eee:insuma:v:86:y:2019:i:c:p:92-97 is not listed on IDEAS
    4. Massimo Marinacci & Luigi Montrucchio, 2005. "Ultramodular Functions," Mathematics of Operations Research, INFORMS, vol. 30(2), pages 311-332, May.
    5. Fernández-Ponce, J.M. & Pellerey, F. & Rodríguez-Griñolo, M.R., 2011. "A characterization of the multivariate excess wealth ordering," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 410-417.
    6. Nicole Bauerle & Alexander Glauner, 2017. "Optimal Risk Allocation in Reinsurance Networks," Papers 1711.10210, arXiv.org.
    7. Müller, Alfred & Scarsini, Marco, 2005. "Archimedean copulæ and positive dependence," Journal of Multivariate Analysis, Elsevier, vol. 93(2), pages 434-445, April.
    8. Xu Guo & Andreas Wagener & Wing-Keung Wong & Lixing Zhu, 2018. "The two-moment decision model with additive risks," Risk Management, Palgrave Macmillan, vol. 20(1), pages 77-94, February.
    9. Charles J. Corbett & Kumar Rajaram, 2006. "A Generalization of the Inventory Pooling Effect to Nonnormal Dependent Demand," Manufacturing & Service Operations Management, INFORMS, vol. 8(4), pages 351-358, August.
    10. Sordo, Miguel A., 2016. "A multivariate extension of the increasing convex order to compare risks," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 224-230.
    11. repec:eee:insuma:v:82:y:2018:i:c:p:37-47 is not listed on IDEAS

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