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Stochastic comparison of random vectors with a common copula


  • Marco Scarsini

    () (GREGH - Groupement de Recherche et d'Etudes en Gestion à HEC - HEC Paris - Ecole des Hautes Etudes Commerciales - CNRS - Centre National de la Recherche Scientifique, Dipartimento di Scienze Economiche e Aziendali - LUISS - Libera Università Internazionale degli Studi Sociali Guido Carli [Roma])

  • Alfred Muller



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

  • Marco Scarsini & Alfred Muller, 2001. "Stochastic comparison of random vectors with a common copula," Post-Print hal-00540198, HAL.
  • Handle: RePEc:hal:journl:hal-00540198
    DOI: 10.1287/moor.26.4.723.10006
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    References listed on IDEAS

    1. Müller, Alfred & Scarsini, Marco, 2000. "Some Remarks on the Supermodular Order," Journal of Multivariate Analysis, Elsevier, vol. 73(1), pages 107-119, April.
    2. Ludolf E. Meester & J. George Shanthikumar, 1999. "Stochastic Convexity on General Space," Mathematics of Operations Research, INFORMS, vol. 24(2), pages 472-494, May.
    3. 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.
    4. Muliere, Pietro & Scarsini, Marco, 1989. "Multivariate decisions with unknown price vector," Economics Letters, Elsevier, vol. 29(1), pages 13-19.
    5. Moshe Shaked & Fabio Spizzichino, 1998. "Positive Dependence Properties of Conditionally Independent Random Lifetimes," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 944-959, November.
    6. Finkelshtain, Israel & Kella, Offer & Scarsini, Marco, 1999. "On risk aversion with two risks," Journal of Mathematical Economics, Elsevier, vol. 31(2), pages 239-250, March.
    7. Machina, Mark J & Pratt, John W, 1997. "Increasing Risk: Some Direct Constructions," Journal of Risk and Uncertainty, Springer, vol. 14(2), pages 103-127, March.
    8. Marco Scarsini, 1998. "Multivariate convex orderings, dependence, and stochastic equality," Post-Print hal-00541775, HAL.
    9. Li, Haijun & Scarsini, Marco & Shaked, Moshe, 1996. "Linkages: A Tool for the Construction of Multivariate Distributions with Given Nonoverlapping Multivariate Marginals," Journal of Multivariate Analysis, Elsevier, vol. 56(1), pages 20-41, January.
    10. Shaked, Moshe & Shanthikumar, J. George, 1997. "Supermodular Stochastic Orders and Positive Dependence of Random Vectors," Journal of Multivariate Analysis, Elsevier, vol. 61(1), pages 86-101, April.
    11. Rüschendorf, Ludger & de Valk, Vincent, 1993. "On regression representations of stochastic processes," Stochastic Processes and their Applications, Elsevier, vol. 46(2), pages 183-198, June.
    12. Rothschild, Michael & Stiglitz, Joseph E., 1970. "Increasing risk: I. A definition," Journal of Economic Theory, Elsevier, vol. 2(3), pages 225-243, September.
    13. Marco Scarsini & Pietro Muliere, 1987. "Characterization of a Marshall-Olkin type class of distributions," Post-Print hal-00542248, HAL.
    14. Reuven Y. Rubinstein & Gennady Samorodnitsky & Moshe Shaked, 1985. "Antithetic Variates, Multivariate Dependence and Simulation of Stochastic Systems," Management Science, INFORMS, vol. 31(1), pages 66-77, January.
    15. Block, Henry W. & Savits, Thomas H. & Shaked, Moshe, 1985. "A concept of negative dependence using stochastic ordering," Statistics & Probability Letters, Elsevier, vol. 3(2), pages 81-86, April.
    16. Marco Scarsini, 1988. "Multivariate stochastic dominance with fixed dependence structure," Post-Print hal-00542234, HAL.
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    Cited by:

    1. Massimo Marinacci & Luigi Montrucchio, 2005. "Ultramodular Functions," Mathematics of Operations Research, INFORMS, vol. 30(2), pages 311-332, May.
    2. Marcello Basili & Paulo Casaca & Alain Chateauneuf & Maurizio Franzini, 2016. "Multidimensional Pigou-Dalton Transfers and Social Evaluation Functions," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01321802, HAL.
    3. 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.
    4. Xu Guo & Andreas Wagener & Wing-Keung Wong & Lixing Zhu, "undated". "The two-moment decision model with additive risks," Risk Management 4, Palgrave Macmillan.
    5. 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.
    6. Nicole Bauerle & Alexander Glauner, 2017. "Optimal Risk Allocation in Reinsurance Networks," Papers 1711.10210,
    7. Müller, Alfred & Scarsini, Marco, 2005. "Archimedean copulæ and positive dependence," Journal of Multivariate Analysis, Elsevier, vol. 93(2), pages 434-445, April.
    8. repec:eee:ejores:v:275:y:2019:i:2:p:755-767 is not listed on IDEAS
    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. repec:eee:insuma:v:86:y:2019:i:c:p:92-97 is not listed on IDEAS
    11. 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.
    12. repec:eee:insuma:v:82:y:2018:i:c:p:37-47 is not listed on IDEAS


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