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ORDERING PROPERTIES OF EXTREME CLAIM AMOUNTS FROM HETEROGENEOUS PORTFOLIOS

Published online by Cambridge University Press:  29 April 2019

Yiying Zhang ,
Xiong Cai  and
Peng Zhao Open the ORCID record for Peng Zhao [Opens in a new window]
Yiying Zhang
Affiliation:
School of Statistics and Data Science, LPMC and KLMDASRNankai UniversityTianjin 300071, P.R. China E-Mail: zhangyiying@outlook.com
Xiong Cai
Affiliation:
College of Applied SciencesBeijing University of TechnologyBeijing 100124, P.R. China E-Mail: xcai@emails.bjut.edu.cn
Peng Zhao*
Affiliation:
School of Mathematics and StatisticsJiangsu Normal UniversityXuzhou 221116, P.R. China E-Mail: zhaop@jsnu.edu.cn
*
E-Mail: zhaop@jsnu.edu.cn
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Abstract

In the context of insurance, the smallest and largest claim amounts turn out to be crucial to insurance analysis since they provide useful information for determining annual premium. In this paper, we establish sufficient conditions for comparing extreme claim amounts arising from two sets of heterogeneous insurance portfolios according to various stochastic orders. It is firstly shown that the weak supermajorization order between the transformed vectors of occurrence probabilities implies the usual stochastic ordering between the largest claim amounts when the claim severities are weakly stochastic arrangement increasing. Secondly, sufficient conditions are established for the right-spread ordering and the convex transform ordering of the smallest claim amounts arising from heterogeneous dependent insurance portfolios with possibly different number of claims. In the setting of independent multiple-outlier claims, we study the effects of heterogeneity among sample sizes on the stochastic properties of the largest and smallest claim amounts in the sense of the hazard rate ordering and the likelihood ratio ordering. Numerical examples are provided to highlight these theoretical results as well. Not only can our results be applied in the area of actuarial science, but also they can be used in other research fields including reliability engineering and auction theory.

Keywords

Largest claim amountssmallest claim amountsstochastic ordersoccurrence probabilitiesmajorization.
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 49 , Issue 2 , May 2019 , pp. 525 - 554
Copyright
Copyright © Astin Bulletin 2019 

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References

Ahmed, A.N., Alzaid, A., Bartoszewicz, J. and Kochar, S.C. (1986) Dispersive and superadditive ordering. Advances in Applied Probability, 18(4), 1019–1022.CrossRefGoogle Scholar
Balakrishnan, N., Zhang, Y. and Zhao, P. (2018) Ordering the largest claim amounts and ranges from two sets of heterogeneous portfolios. Scandinavian Actuarial Journal, 2018(1), 23–41.CrossRefGoogle Scholar
Barmalzan, G. and Payandeh Najafabadi, A.T. (2015) On the convex transform and right-spread orders of smallest claim amounts. Insurance: Mathematics and Economics, 64, 380–384.Google Scholar
Barmalzan, G., Payandeh Najafabadi, A.T. and Balakrishnan, N. (2015) Stochastic comparison of aggregate claim amounts between two heterogeneous portfolios and its applications. Insurance: Mathematics and Economics, 61, 235–241.Google Scholar
Barmalzan, G., Payandeh Najafabadi, A.T. and Balakrishnan, N. (2016) Likelihood ratio and dispersive orders for smallest order statistics and smallest claim amounts from heterogeneous Weibull sample. Statistics & Probability Letters, 110, 1–7.CrossRefGoogle Scholar
Barmalzan, G., Payandeh Najafabadi, A.T. and Balakrishnan, N. (2017) Ordering properties of the smallest and largest claim amounts in a general scale model. Scandinavian Actuarial Journal, 2017(2), 105–124.CrossRefGoogle Scholar
Bartoszewicz, J. (1985) Dispersive ordering and monotone failure rate distributions. Advances in Applied Probability, 17(2), 472–474.CrossRefGoogle Scholar
Bühlmann, H. (1980) An economic premium principle. ASTIN Bulletin, 11(1), 52–60.CrossRefGoogle Scholar
Cai, J. and Wei, W. (2014) Some new notions of dependence with applications in optimal allocation problems. Insurance: Mathematics and Economics, 55, 200–209.Google Scholar
Cai, J. and Wei, W. (2015) Notions of multivariate dependence and their applications in optimal portfolio selections with dependent risks. Journal of Multivariate Analysis, 138, 156–169.CrossRefGoogle Scholar
Denuit, M., Dhaene, J., Goovaerts, M.J. and Kaas, R. (2006). Actuarial Theory for Dependent Risks: Measures, Orders and Models. John Wiley & Sons.Google Scholar
Denuit, M. and Frostig, E. (2006) Heterogeneity and the need for capital in the individual model. Scandinavian Actuarial Journal, 2006(1), 42–66.CrossRefGoogle Scholar
Frostig, E. (2001). A comparison between homogeneous and heterogeneous portfolios. Insurance: Mathematics and Economics, 29, 59-71.Google Scholar
Hua, L. and Cheung, K.C. (2008) Stochastic orders of scalar products with applications. Insurance: Mathematics and Economics, 42(3), 865–872.Google Scholar
Khaledi, B.E. and Ahmadi, S.S. (2008) On stochastic comparison between aggregate claim amounts. Journal of Statistical Planning and Inference, 138(7), 3121–3129.CrossRefGoogle Scholar
Kochar, S.C. and Xu, M. (2009) Comparisons of parallel systems according to the convex transform order. Journal of Applied Probability, 46(2), 342–352.CrossRefGoogle Scholar
Kochar, S.C. and Xu, M. (2010) On the right spread order of convolutions of heterogeneous exponential random variables. Journal of Multivariate Analysis, 101(1), 165–176.CrossRefGoogle Scholar
Kochar, S.C., Li, X. and Shaked, M. (2002) The total time on test transform and the excess wealth stochastic orders of distributions. Advances in Applied Probability, 34(4), 826–845.CrossRefGoogle Scholar
Ma, C. (2000) Convex orders for linear combinations of random variables. Journal of Statistical Planning and Inference, 84(1), 11–25.CrossRefGoogle Scholar
Marshall, A.W., Olkin, I. and Arnold, B.C. (2011) Inequalities: Theory of Majorization and Its Applications, 2nd ed. New York: Springer-Verlag.CrossRefGoogle Scholar
Murthy, D.N.P., Xie, M. and Jiang, R. (2004) Weibull Models. New Jersey: John Wiley & Sons.Google Scholar
Müller, A. and Stoyan, D. (2002) Comparison Methods for Stochastic Models and Risks. New York: John Wiley & Sons.Google Scholar
Nelsen, R.B. (2006) An Introduction to Copulas. New York: Springer.Google Scholar
Shaked, M. and Shanthikumar, J.G. (2007) Stochastic Orders. New York: Springer-Verlag.CrossRefGoogle Scholar
Van Heerwaarden, A.E., Kaas, R. and Goovaerts, M.J. (1989) Properties of the Esscher premium calculation principle. Insurance: Mathematics and Economics, 8, 261–267.Google Scholar
Van Zwet, W.R. (1970) Convex Transformations of Random Variables (Math. Centre Tracts 7), 2nd ed. Mathematical Centre: Amsterdam.Google Scholar
Wang, S. (1996) Premium calculation by transforming the layer premium density. ASTIN Bulletin, 26(1), 71–92.CrossRefGoogle Scholar
Yaari, M.E. (1987) The dual theory of choice under risk. Econometrica, 55(1), 95–115.CrossRefGoogle Scholar
Zhang, Y. and Zhao, P. (2015) Comparisons on aggregate risks from two sets of heterogeneous portfolios. Insurance: Mathematics and Economics, 65, 124–135.Google Scholar
Zhang, Y., Amini-Seresht, E. and Zhao, P. (2018a) On fail-safe systems under random shocks. Applied Stochastic Models in Business and Industry, https://doi.org/10.1002/asmb.2349.CrossRefGoogle Scholar
Zhang, Y., Li, X. and Cheung, K.C. (2018b) On heterogeneity in the individual model with both dependent claim occurrences and severities. ASTIN Bulletin, 48(2), 817–839.CrossRefGoogle Scholar