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Multivariate convex orderings, dependence, and stochastic equality

Author

Listed:
  • 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])

Abstract

We consider the convex ordering for random vectors and some weaker versions of it, like the convex ordering for linear combinations of random variables. First we establish conditions of stochastic equality for random vectors that are ordered by one of the convex orderings. Then we establish necessary and sufficient conditions for the convex ordering to hold in the case of multivariate normal distributions and sufficient conditions for the positive linear convex ordering (without the restriction to multi-normality).

Suggested Citation

  • Marco Scarsini, 1998. "Multivariate convex orderings, dependence, and stochastic equality," Post-Print hal-00541775, HAL.
    • Handle: RePEc:hal:journl:hal-00541775
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    Citations

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    Cited by:

    1. Belzunce, Félix & Suárez-Llorens, Alfonso & Sordo, Miguel A., 2012. "Comparison of increasing directionally convex transformations of random vectors with a common copula," Insurance: Mathematics and Economics, Elsevier, vol. 50(3), pages 385-390.
    2. Georg Ch. Pflug & Alois Pichler, 2018. "Systemic risk and copula models," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 26(2), pages 465-483, June.
    3. Fernández, Ignacio Cascos & Molchanov, Ilya, 2003. "A stochastic order for random vectors and random sets based on the Aumann expectation," Statistics & Probability Letters, Elsevier, vol. 63(3), pages 295-305, July.
    4. Arlotto, Alessandro & Scarsini, Marco, 2009. "Hessian orders and multinormal distributions," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2324-2330, November.
    5. 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.
    6. Chuancun Yin, 2019. "Stochastic Orderings of Multivariate Elliptical Distributions," Papers 1910.07158, arXiv.org, revised Nov 2019.
    7. Julien Guyon & Romain Menegaux & Marcel Nutz, 2017. "Bounds for VIX futures given S&P 500 smiles," Finance and Stochastics, Springer, vol. 21(3), pages 593-630, July.
    8. Pan, Xiaoqing & Qiu, Guoxin & Hu, Taizhong, 2016. "Stochastic orderings for elliptical random vectors," Journal of Multivariate Analysis, Elsevier, vol. 148(C), pages 83-88.
    9. Miguel Sordo & Héctor Ramos, 2007. "Characterization of stochastic orders by L-functionals," Statistical Papers, Springer, vol. 48(2), pages 249-263, April.
    10. Amiri, Mehdi & Izadkhah, Salman & Jamalizadeh, Ahad, 2020. "Linear orderings of the scale mixtures of the multivariate skew-normal distribution," Journal of Multivariate Analysis, Elsevier, vol. 179(C).
    11. Julien Guyon & Romain Menegaux & Marcel Nutz, 2016. "Bounds for VIX Futures given S&P 500 Smiles," Papers 1609.05832, arXiv.org, revised Jun 2017.
    12. Diwakar Gupta & William L. Cooper, 2005. "Stochastic Comparisons in Production Yield Management," Operations Research, INFORMS, vol. 53(2), pages 377-384, April.
    13. Mehdi Amiri & Narayanaswamy Balakrishnan & Abbas Eftekharian, 2022. "Hessian orderings of multivariate normal variance-mean mixture distributions and their applications in evaluating dependent multivariate risk portfolios," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 31(3), pages 679-707, September.
    14. Francesco Andreoli & Claudio Zoli, 2020. "From unidimensional to multidimensional inequality: a review," METRON, Springer;Sapienza Università di Roma, vol. 78(1), pages 5-42, April.
    15. Alfred Müller, 2001. "Stochastic Ordering of Multivariate Normal Distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(3), pages 567-575, September.
    16. Marco Dall’Aglio & Marco Scarsini, 2000. "Zonoids, Linear Dependence, and Size-Biased Distributions on the Simplex," ICER Working Papers - Applied Mathematics Series 27-2003, ICER - International Centre for Economic Research, revised Jul 2003.
    17. 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.
    18. Ebru K. Bish & Juqi Liu & Douglas R. Bish, 2010. "A note on “resource flexibility with responsive pricing”," Naval Research Logistics (NRL), John Wiley & Sons, vol. 57(2), pages 119-126, March.
    19. Denuit, Michel & Lefèvre, Claude & Shaked, Moshe, 2000. "On the theory of high convexity stochastic orders," Statistics & Probability Letters, Elsevier, vol. 47(3), pages 287-293, April.

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