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Stochastic orders of scalar products with applications

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  • Hua, Lei
  • Cheung, Ka Chun

Abstract

In this paper, we study stochastic orders of scalar products of random vectors. Based on the study of Ma [Ma, C., 2000. Convex orders for linear combinations of random variables. J. Statist. Plann. Inference 84, 11-25], we first obtain more general conditions under which linear combinations of random variables can be ordered in the increasing convex order. As an application of this result, we consider the scalar product of two random vectors which separates the severity effect and the frequency effect in the study of the optimal allocation of policy limits and deductibles. Finally, we obtain the ordering of the optimal allocation of policy limits and deductibles when the dependence structure of the losses is unknown. This application is a further study of Cheung [Cheung, K.C., 2007. Optimal allocation of policy limits and deductibles. Insurance: Math. Econom. 41, 382-391].

Suggested Citation

  • Hua, Lei & Cheung, Ka Chun, 2008. "Stochastic orders of scalar products with applications," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 865-872, June.
    • Handle: RePEc:eee:insuma:v:42:y:2008:i:3:p:865-872
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    References listed on IDEAS

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    1. 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.
    2. Cheung, Ka Chun, 2006. "Optimal portfolio problem with unknown dependency structure," Insurance: Mathematics and Economics, Elsevier, vol. 38(1), pages 167-175, February.
    3. Cheung, Ka Chun, 2007. "Optimal allocation of policy limits and deductibles," Insurance: Mathematics and Economics, Elsevier, vol. 41(3), pages 382-391, November.
    4. Kaas, Rob & Dhaene, Jan & Goovaerts, Marc J., 2000. "Upper and lower bounds for sums of random variables," Insurance: Mathematics and Economics, Elsevier, vol. 27(2), pages 151-168, October.
    5. Dhaene, Jan & Goovaerts, Marc J., 1996. "Dependency of Risks and Stop-Loss Order1," ASTIN Bulletin, Cambridge University Press, vol. 26(2), pages 201-212, November.
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    Cited by:

    1. Manesh, Sirous Fathi & Khaledi, Baha-Eldin & Dhaene, Jan, 2016. "Optimal allocation of policy deductibles for exchangeable risks," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 87-92.
    2. Zhuang, Weiwei & Chen, Zijin & Hu, Taizhong, 2009. "Optimal allocation of policy limits and deductibles under distortion risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 44(3), pages 409-414, June.
    3. Manesh, Sirous Fathi & Khaledi, Baha-Eldin, 2015. "Allocations of policy limits and ordering relations for aggregate remaining claims," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 9-14.
    4. Maria Mercè Claramunt & Maite Màrmol, 2020. "Refundable deductible insurance," Working Papers hal-02909299, HAL.
    5. Laureano Escudero & Eva-María Ortega, 2009. "How retention levels influence the variability of the total risk under reinsurance," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 17(1), pages 139-157, July.
    6. Yinping You & Xiaohu Li, 2017. "Most unfavorable deductibles and coverage limits for multiple random risks with Archimedean copulas," Annals of Operations Research, Springer, vol. 259(1), pages 485-501, December.
    7. Xiaohu Li & Yinping You, 2014. "A note on allocation of portfolio shares of random assets with Archimedean copula," Annals of Operations Research, Springer, vol. 212(1), pages 155-167, January.
    8. Cheung, Ka Chun, 2009. "Applications of conditional comonotonicity to some optimization problems," Insurance: Mathematics and Economics, Elsevier, vol. 45(1), pages 89-93, August.
    9. Li, Chen & Li, Xiaohu, 2017. "Ordering optimal deductible allocations for stochastic arrangement increasing risks," Insurance: Mathematics and Economics, Elsevier, vol. 73(C), pages 31-40.
    10. Franco Pellerey & Saeed Zalzadeh, 2015. "A note on relationships between some univariate stochastic orders and the corresponding joint stochastic orders," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(4), pages 399-414, May.
    11. Wei, Wei, 2017. "Joint stochastic orders of high degrees and their applications in portfolio selections," Insurance: Mathematics and Economics, Elsevier, vol. 76(C), pages 141-148.
    12. Yinping You & Xiaohu Li & Rui Fang, 2021. "On coverage limits and deductibles for SAI loss severities," Annals of Operations Research, Springer, vol. 297(1), pages 341-357, February.
    13. Zhang, Yiying & Cheung, Ka Chun, 2020. "On the increasing convex order of generalized aggregation of dependent random variables," Insurance: Mathematics and Economics, Elsevier, vol. 92(C), pages 61-69.
    14. Halim Zeghdoudi & Meriem Bouhadjar & Mohamed Riad Remita, 2014. "On Stochastic Orders and its applications : Policy limits and Deductibles," Papers 1411.1609, arXiv.org, revised Jan 2015.
    15. Li, Xiaohu & You, Yinping, 2012. "On allocation of upper limits and deductibles with dependent frequencies and comonotonic severities," Insurance: Mathematics and Economics, Elsevier, vol. 50(3), pages 423-429.
    16. Lu, ZhiYi & Meng, LiLi, 2011. "Stochastic comparisons for allocations of policy limits and deductibles with applications," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 338-343, May.

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