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Stochastic comparisons of distorted variability measures

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  • Sordo, Miguel A.
  • Suárez-Llorens, Alfonso
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    Abstract

    In this paper, we consider the dispersive order and the excess wealth order to compare the variability of distorted distributions. We know from Sordo (2009a) that the excess wealth order can be characterized in terms of a class of variability measures associated to the tail conditional distribution which includes, as a particular measure, the tail variance. Given that the tail conditional distribution is a particular distorted distribution, a natural question is whether this result can be extended to include other classes of variability measures associated to general distorted distributions. As we show in this paper, the answer is yes, by focusing on distorted distributions associated to concave distortion functions. For distorted distributions associated to more general distortions, the characterizations are stated in terms of the stronger dispersive order.

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    Bibliographic Info

    Article provided by Elsevier in its journal Insurance: Mathematics and Economics.

    Volume (Year): 49 (2011)
    Issue (Month): 1 (July)
    Pages: 11-17

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    Handle: RePEc:eee:insuma:v:49:y:2011:i:1:p:11-17

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    Web page: http://www.elsevier.com/locate/inca/505554

    Related research

    Keywords: Excess wealth order Dispersive order Distorted distributions Distortions Tail variance;

    References

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    1. Sordo, Miguel A., 2008. "Characterizations of classes of risk measures by dispersive orders," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 42(3), pages 1028-1034, June.
    2. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, Econometric Society, vol. 57(3), pages 571-87, May.
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    4. Furman, Edward & Zitikis, Ricardas, 2008. "Weighted premium calculation principles," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 42(1), pages 459-465, February.
    5. Goovaerts, Marc J. & Laeven, Roger J.A., 2008. "Actuarial risk measures for financial derivative pricing," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 42(2), pages 540-547, April.
    6. Sordo, Miguel A., 2009. "Comparing tail variabilities of risks by means of the excess wealth order," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 45(3), pages 466-469, December.
    7. Hu, Taizhong & Chen, Jing & Yao, Junchao, 2006. "Preservation of the location independent risk order under convolution," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 38(2), pages 406-412, April.
    8. Shaun, Wang, 1995. "Insurance pricing and increased limits ratemaking by proportional hazards transforms," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 17(1), pages 43-54, August.
    9. Chateauneuf, Alain & Cohen, Michele & Meilijson, Isaac, 2004. "Four notions of mean-preserving increase in risk, risk attitudes and applications to the rank-dependent expected utility model," Journal of Mathematical Economics, Elsevier, Elsevier, vol. 40(5), pages 547-571, August.
    10. Sordo, Miguel A., 2009. "On the relationship of location-independent riskier order to the usual stochastic order," Statistics & Probability Letters, Elsevier, Elsevier, vol. 79(2), pages 155-157, January.
    11. Goovaerts, Marc J. & Kaas, Rob & Laeven, Roger J.A., 2010. "Decision principles derived from risk measures," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 47(3), pages 294-302, December.
    12. Rojo, Javier & He, Guo Zhong, 1991. "New properties and characterizations of the dispersive ordering," Statistics & Probability Letters, Elsevier, Elsevier, vol. 11(4), pages 365-372, April.
    13. Quiggin, John, 1982. "A theory of anticipated utility," Journal of Economic Behavior & Organization, Elsevier, Elsevier, vol. 3(4), pages 323-343, December.
    14. Jones, Bruce L. & Zitikis, Ricardas, 2007. "Risk measures, distortion parameters, and their empirical estimation," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 41(2), pages 279-297, September.
    15. Yaari, Menahem E, 1987. "The Dual Theory of Choice under Risk," Econometrica, Econometric Society, Econometric Society, vol. 55(1), pages 95-115, January.
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    Cited by:
    1. Jaume Belles-Sampera & José M. Merigó & Montserrat Guillén & Miguel Santolino, 2012. "The connection between distortion risk measures and ordered weighted averaging operators," IREA Working Papers, University of Barcelona, Research Institute of Applied Economics 201201, University of Barcelona, Research Institute of Applied Economics, revised Jan 2012.
    2. Belzunce, Félix & Pinar, José F. & Ruiz, José M. & Sordo, Miguel A., 2012. "Comparison of risks based on the expected proportional shortfall," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 51(2), pages 292-302.
    3. López-Díaz, Miguel & Sordo, Miguel A. & Suárez-Llorens, Alfonso, 2012. "On the Lp-metric between a probability distribution and its distortion," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 51(2), pages 257-264.
    4. Gómez-Déniz, Emilio & Sordo, Miguel A. & Calderín-Ojeda, Enrique, 2014. "The Log–Lindley distribution as an alternative to the beta regression model with applications in insurance," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 54(C), pages 49-57.
    5. 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, Elsevier, vol. 50(3), pages 385-390.

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