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Characterizing a comonotonic random vector by the distribution of the sum of its components

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  • Cheung, Ka Chun

Abstract

In this article, we characterize comonotonicity and related dependence structures among several random variables by the distribution of their sum. First we prove that if the sum has the same distribution as the corresponding comonotonic sum, then the underlying random variables must be comonotonic as long as each of them is integrable. In the literature, this result is only known to be true if either each random variable is square integrable or possesses a continuous distribution function. We then study the situation when the distribution of the sum only coincides with the corresponding comonotonic sum in the tail. This leads to the dependence structure known as tail comonotonicity. Finally, by establishing some new results concerning convex order, we show that comonotonicity can also be characterized by expected utility and distortion risk measures.

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  • Cheung, Ka Chun, 2010. "Characterizing a comonotonic random vector by the distribution of the sum of its components," Insurance: Mathematics and Economics, Elsevier, vol. 47(2), pages 130-136, October.
    • Handle: RePEc:eee:insuma:v:47:y:2010:i:2:p:130-136
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    References listed on IDEAS

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    1. Muller, Alfred, 1996. "Orderings of risks: A comparative study via stop-loss transforms," Insurance: Mathematics and Economics, Elsevier, vol. 17(3), pages 215-222, April.
    2. Cheung, Ka Chun, 2010. "Comonotonic convex upper bound and majorization," Insurance: Mathematics and Economics, Elsevier, vol. 47(2), pages 154-158, October.
    3. Cheung, Ka Chun, 2008. "Characterization of comonotonicity using convex order," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 403-406, December.
    4. Denuit Michel & Dhaene Jan & Goovaerts Marc & Kaas Rob & Laeven Roger, 2006. "Risk measurement with equivalent utility principles," Statistics & Risk Modeling, De Gruyter, vol. 24(1), pages 1-25, July.
    5. Dhaene, Jan & Denuit, Michel & Vanduffel, Steven, 2009. "Correlation order, merging and diversification," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 325-332, December.
    6. Dhaene, J. & Denuit, M. & Goovaerts, M. J. & Kaas, R. & Vyncke, D., 2002. "The concept of comonotonicity in actuarial science and finance: theory," Insurance: Mathematics and Economics, Elsevier, vol. 31(1), pages 3-33, August.
    7. Dhaene, J. & Denuit, M. & Goovaerts, M. J. & Kaas, R. & Vyncke, D., 2002. "The concept of comonotonicity in actuarial science and finance: applications," Insurance: Mathematics and Economics, Elsevier, vol. 31(2), pages 133-161, October.
    8. Denuit Michel & Dhaene Jan & Goovaerts Marc & Kaas Rob & Laeven Roger, 2006. "Risk measurement with equivalent utility principles," Statistics & Risk Modeling, De Gruyter, vol. 24(1), pages 1-25, July.
    9. Cheung, Ka Chun, 2009. "Upper comonotonicity," Insurance: Mathematics and Economics, Elsevier, vol. 45(1), pages 35-40, August.
    10. Kaas, R. & Dhaene, J. & Vyncke, D. & Goovaerts, M.J. & Denuit, M., 2002. "A Simple Geometric Proof that Comonotonic Risks Have the Convex-Largest Sum," ASTIN Bulletin, Cambridge University Press, vol. 32(1), pages 71-80, May.
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    Cited by:

    1. Cheung, Ka Chun & Lo, Ambrose, 2013. "Characterizations of counter-monotonicity and upper comonotonicity by (tail) convex order," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 334-342.
    2. Mao, Tiantian & Hu, Taizhong, 2011. "A new proof of Cheung's characterization of comonotonicity," Insurance: Mathematics and Economics, Elsevier, vol. 48(2), pages 214-216, March.
    3. Dhaene, Jan & Linders, Daniël & Schoutens, Wim & Vyncke, David, 2012. "The Herd Behavior Index: A new measure for the implied degree of co-movement in stock markets," Insurance: Mathematics and Economics, Elsevier, vol. 50(3), pages 357-370.
    4. Cheung, Ka Chun & Dhaene, Jan & Lo, Ambrose & Tang, Qihe, 2014. "Reducing risk by merging counter-monotonic risks," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 58-65.
    5. Nam, Hee Seok & Tang, Qihe & Yang, Fan, 2011. "Characterization of upper comonotonicity via tail convex order," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 368-373, May.
    6. Cheung, Ka Chun & Lo, Ambrose, 2013. "General lower bounds on convex functionals of aggregate sums," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 884-896.
    7. Wang, Ruodu & Zitikis, Ričardas, 2020. "Weak comonotonicity," European Journal of Operational Research, Elsevier, vol. 282(1), pages 386-397.
    8. Cheung, Ka Chun & Lo, Ambrose, 2014. "Characterizing mutual exclusivity as the strongest negative multivariate dependence structure," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 180-190.

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