Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-24T17:34:55.585Z Has data issue: false hasContentIssue false
  • English
  • Français

On transform orders for largest claim amounts

Part of: Distribution theory - Probability Mathematical economics

Published online by Cambridge University Press:  22 November 2021

Yiying Zhang Open the ORCID record for Yiying Zhang [Opens in a new window]
Yiying Zhang*
Affiliation:
Southern University of Science and Technology
*
*Postal address: Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, Guangdong, China. Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper investigates the ordering properties of largest claim amounts in heterogeneous insurance portfolios in the sense of some transform orders, including the convex transform order and the star order. It is shown that the largest claim amount from a set of independent and heterogeneous exponential claims is more skewed than that from a set of independent and homogeneous exponential claims in the sense of the convex transform order. As a result, a lower bound for the coefficient of variation of the largest claim amount is established without any restrictions on the parameters of the distributions of claim severities. Furthermore, sufficient conditions are presented to compare the skewness of the largest claim amounts from two sets of independent multiple-outlier scaled claims according to the star order. Some comparison results are also developed for the multiple-outlier proportional hazard rates claims. Numerical examples are presented to illustrate these theoretical results.

Keywords

Largest claim amountskewnessstochastic ordersmajorizationscale modelPHR model

MSC classification

Primary: 91B16: Utility theory 91B30: Risk theory, insurance
Secondary: 60E15: Inequalities; stochastic orderings
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 4 , December 2021 , pp. 1064 - 1085
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahmed, A. N., Alzaid, A., Bartoszewicz, J. and Kochar, S. C. (1986). Dispersive and superadditive ordering. Adv. Appl. Prob. 18, 1019–1022.10.2307/1427262CrossRefGoogle Scholar
Amini-Seresht, E., Qiao, J., Zhang, Y. and Zhao, P. (2016). On the skewness of order statistics in multiple-outlier PHR models. Metrika 79, 817–836.10.1007/s00184-016-0579-7CrossRefGoogle Scholar
Balakrishnan, N., Zhang, Y. and Zhao, P. (2018). Ordering the largest claim amounts and ranges from two sets of heterogeneous portfolios. Scand. Actuarial J. 2018, 23–41.10.1080/03461238.2017.1278717CrossRefGoogle Scholar
Barlow, R. and Proschan, R. (1981). Statistical Theory of Reliability and Life Testing. Silver Spring, MD: Madison.Google Scholar
Barmalzan, G. and Payandeh Najafabadi, A. T. (2015). On the convex transform and right-spread orders of smallest claim amounts. Insurance Math. Econom. 64, 380–384.10.1016/j.insmatheco.2015.07.001CrossRefGoogle 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 Math. Econom. 61, 235–241.10.1016/j.insmatheco.2015.01.010CrossRefGoogle 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. Statist. Prob. Lett. 110, 1–7.10.1016/j.spl.2015.11.009CrossRefGoogle 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. Scand. Actuarial J. 2017, 105–124.10.1080/03461238.2015.1090476CrossRefGoogle Scholar
Benktander, G. (1978). Largest claims reinsurance (LCR). A quick method to calculate LCR-risk rates from excess of loss risk rates. ASTIN Bull. 10, 54–58.10.1017/S0515036100006346CrossRefGoogle Scholar
Denuit, M., Dhaene, J., Goovaerts, M. J. and Kaas, R. (2006). Actuarial Theory for Dependent Risks: Measures, Orders and Models. John Wiley, Chichester.Google Scholar
Denuit, M. and Frostig, E. (2006). Heterogeneity and the need for capital in the individual model. Scand. Actuarial J. 2006, 42–66.10.1080/03461230500518690CrossRefGoogle Scholar
Dhaene, J. and Goovaerts, M. J. (1996). Dependency of risks and stop-loss order. ASTIN Bull. 26, 201–212.10.2143/AST.26.2.563219CrossRefGoogle Scholar
Dhaene, J. and Goovaerts, M. J. (1997). On the dependency of risks in the individual life model. Insurance Math. Econom. 19, 243–253.10.1016/S0167-6687(96)00015-7CrossRefGoogle Scholar
Ding, W., Yang, J. and Ling, X. (2017). On the skewness of extreme order statistics from heterogeneous samples. Commun. Statist. Theory Meth. 46, 2315–2331.10.1080/03610926.2015.1041984CrossRefGoogle Scholar
Fernández-Ponce, J. M., Kochar, S. C. and Muñoz-Perez, J. (1998). Partial orderings of distributions based on right-spread functions. J. Appl. Prob. 35, 221–228.10.1239/jap/1032192565CrossRefGoogle Scholar
Frostig, E. (2001). A comparison between homogeneous and heterogeneous portfolios. Insurance Math. Econom. 29, 59–71.10.1016/S0167-6687(01)00073-7CrossRefGoogle Scholar
Frostig, E. (2006). On risk dependence and MRL ordering. Statist. Prob. Lett. 76, 231–243.10.1016/j.spl.2005.07.012CrossRefGoogle Scholar
Hu, T. and Ruan, L. (2004). A note on multivariate stochastic comparisons of Bernoulli random variable. J. Statist. Planning Infer. 126, 281–288.10.1016/j.jspi.2003.07.012CrossRefGoogle Scholar
Hu, T. and Wu, Z. (1999). On dependence of risks and stop-loss premiums. Insurance Math. Econom. 24, 323–332.10.1016/S0167-6687(99)00007-4CrossRefGoogle Scholar
Khaledi, B. E. and Ahmadi, S. S. (2008). On stochastic comparison between aggregate claim amounts. J. Statist. Planning Infer. 138, 2243–2251.10.1016/j.jspi.2007.10.018CrossRefGoogle Scholar
Khaledi, B. E., Farsinezhad, S. and Kochar, S. C. (2011). Stochastic comparisons of order statistics in the scale models. J. Statist. Planning Infer. 141, 276–286.10.1016/j.jspi.2010.06.006CrossRefGoogle Scholar
Kochar, S. C., Li, X. and Shaked, M. (2002). The total time on test transform and the excess wealth stochastic orders of distributions. Adv. Appl. Prob. 34, 826–845.10.1239/aap/1037990955CrossRefGoogle Scholar
Kochar, S. C. and Xu, M. (2009). Comparisons of parallel systems according to the convex transform order. J. Appl. Prob. 46, 342–352.10.1017/S0021900200005490CrossRefGoogle Scholar
Kochar, S. C. and Xu, M. (2010). On the right spread order of convolutions of heterogeneous exponential random variables. J. Multivar. Anal. 101, 165–176.10.1016/j.jmva.2009.07.001CrossRefGoogle Scholar
Kochar, S. C. and Xu, M. (2014). On the skewness of order statistics with applications. Ann. Operat. Res. 212, 127–138.10.1007/s10479-012-1212-4CrossRefGoogle Scholar
Kremer, E. (1983). Distribution-free upper bounds on the premiums of the LCR and ECOMOR treaties. Insurance Math. Econom. 2, 209–213.10.1016/0167-6687(83)90013-6CrossRefGoogle Scholar
Kremer, E. (1998). Largest claims reinsurance premiums under possible claims dependence. ASTIN Bull. 28, 257–267.10.2143/AST.28.2.519069CrossRefGoogle Scholar
Ladoucette, S. A. and Teugels, J. L. (2006). Reinsurance of large claims. J. Comput. Appl. Math. 186, 163–190.10.1016/j.cam.2005.03.069CrossRefGoogle Scholar
Ma, C. (2000). Convex orders for linear combinations of random variables. J. Statist. Planning Infer. 84, 11–25.10.1016/S0378-3758(99)00143-3CrossRefGoogle Scholar
Marshall, A. W. and Olkin, I. (2007). Life Distributions. Springer, New York.Google Scholar
Marshall, A. W., Olkin, I. and Arnold, B. C. (2011). Inequalities: Theory of Majorization and Its Applications, 2nd edn. Springer, New York.10.1007/978-0-387-68276-1CrossRefGoogle Scholar
Mitrinović, D. S. (1970). Analytic Inequalities. Springer, Berlin.10.1007/978-3-642-99970-3CrossRefGoogle Scholar
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, New York.Google Scholar
Saunders, I. W. and Moran, P. A. (1978). On the quantiles of the gamma and F distributions. J. Appl. Prob. 15, 426–432.10.2307/3213414CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.10.1007/978-0-387-34675-5CrossRefGoogle Scholar
Van Zwet, W. R. (1970). Convex Transformations of Random Variables (Math. Centre Tracts 7), 2nd edn. Mathematical Centre, Amsterdam.Google Scholar
Wang, Y. (1993). On the number of successes in independent trials. Statistica Sinica 3, 295–312.Google Scholar
Xu, M. and Balakrishnan, N. (2011). On the convolution of heterogeneous Bernoulli random variables. J. Appl. Prob. 48, 877–884.10.1239/jap/1316796922CrossRefGoogle Scholar
Zhang, Y. and Zhao, P. (2015). Comparisons on aggregate risks from two sets of heterogeneous portfolios. Insurance Math. Econom. 65, 124–135.10.1016/j.insmatheco.2015.09.004CrossRefGoogle Scholar
Zhang, Y., Amini-Seresht, E. and Zhao, P. (2019a). On fail-safe systems under random shocks. Appl. Stoch. Models Business Industry 35, 591–602.10.1002/asmb.2349CrossRefGoogle Scholar
Zhang, Y., Cai, X. and Zhao, P. (2019b). Ordering properties of extreme claim amounts from heterogeneous portfolios. ASTIN Bull. 49, 525–554.10.1017/asb.2019.7CrossRefGoogle Scholar
Zhang, Y., Zhao, P. and Cheung, K. C. (2019c). Comparisons on aggregate claim numbers and amount: A study of heterogeneity. Scand. Actuarial J. 2019, 273–290.10.1080/03461238.2018.1557738CrossRefGoogle Scholar
Zhang, Y., Ding, W. and Zhao, P. (2020). On variability of series and parallel systems with heterogeneous components. Prob. Eng. Inf. Sci. 34, 626–645.10.1017/S0269964819000263CrossRefGoogle Scholar