Hostname: page-component-745bb68f8f-mzp66 Total loading time: 0 Render date: 2025-01-10T20:35:04.206Z Has data issue: false hasContentIssue false
  • English
  • Français

ON SUMS OF INDEPENDENT GENERALIZED PARETO RANDOM VARIABLES WITH APPLICATIONS TO INSURANCE AND CAT BONDS

Published online by Cambridge University Press:  04 April 2017

Saralees Nadarajah ,
Yuanyuan Zhang  and
Tibor K. Pogány
Saralees Nadarajah
Affiliation:
School of Mathematics, University of Manchester, Manchester, UK E-mail: [email protected]
Yuanyuan Zhang
Affiliation:
School of Mathematics, University of Manchester, Manchester, UK
Tibor K. Pogány
Affiliation:
Faculty of Maritime Studies, University of Rijeka, Rijeka, CROATIA
Get access Rights & Permissions [Opens in a new window]

Abstract

We derive single integral representations for the exact distribution of the sum of independent generalized Pareto random variables. The integrands involve the incomplete and complementary incomplete gamma functions. Applications to insurance and catastrophe bonds are described.

Keywords

characteristic functiongeneralized Pareto distributionincomplete gamma function
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 32 , Issue 2 , April 2018 , pp. 296 - 305
Copyright
Copyright © Cambridge University Press 2017 

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

1. Abate, J. & Valkó, P.P. (2004). Multi-precision Laplace transform inversion. International Journal for Numerical Methods in Engineering, 60: 979–993.Google Scholar
2. Albrecher, H. & Kortschak, D. (2009). On ruin probability and aggregate claim representations for Pareto claim size distributions. Insurance: Mathematics and Economics, 45: 362–373.Google Scholar
3. Amari, S.V. & Misra, R.B. (1997). Closed-form expressions for distribution of sum of exponential random variables. IEEE Transactions on Reliability, 46: 519–522.Google Scholar
4. Bazargan, H., Bahai, H., & Aminzadeh-Gohari, A. (2007). Calculating the return value using a mathematical model of significant wave height. Journal of Marine Science and Technology, 12: 34–42.Google Scholar
5. Bean, M.A. (2001). Probability: The science of uncertainty: with applications to investments, insurance, and engineering. Providence, RI: American Mathematical Society.Google Scholar
6. Bonfiglioli, A. & Gancia, G. (2013). Heterogeneity, selection and labor market disparities. Working Paper 734, Barcelona Graduate School of Economics, Spain.Google Scholar
7. Goovaerts, M.J., Kaas, R., Laeven, R.J.A., Tang, Q., & Vernic, R. (2005). The tail probability of discounted sums of Pareto-like losses in insurance. Scandinavian Actuarial Journal, 6: 446–461.Google Scholar
8. Hempel, C.G. (2007). Track initialization for multi-static active sonar systems. OCEANS 2007 – Europe, pp. 1–6.Google Scholar
9. Hitha, N. (1991). Some characterizations of Pareto and related populations. Ph.D. thesis, Department of Mathematics and Statistics, Cochin University of Science and Technology, Kochi, India.Google Scholar
10. Khuong, H.V. & Kong, H.-Y. (2006). General expression for pdf of a sum of independent exponential random variables. IEEE Communications Letters, 10: 159–161.Google Scholar
11. Klugman, S.A., Panjer, H.H., & Willmot, G.E. (2008). Loss models, 3rd ed. Hoboken, New Jersey: John Wiley and Sons.CrossRefGoogle Scholar
12. Morales, M. (2005). On an approximation for the surplus process using extreme value theory: Applications in ruin theory and reinsurance pricing. North American Actuarial Journal, 8: 46–66.Google Scholar
13. Nadarajah, S. (2008). Generalized Pareto models with application to drought data. Environmetrics, 19: 395–408.CrossRefGoogle Scholar
14. Nadarajah, S. & Kotz, S. (2006). On the Laplace transform of the Pareto distribution. Queueing Systems, 54: 243–244.Google Scholar
15. Nadarajah, S. & Pogány, T.K. (2013). On the characteristic functions for extreme value distributions. Extremes, 16: 27–38.Google Scholar
16. Nguyen, Q.H. & Robert, C. (2015). Series expansions for convolutions of Pareto distributions. Statistics and Risk Modeling, 32: doi:10.1515/strm-2014-1168 Google Scholar
17. Pareto, V. (1964). Cours d'Économie Politique: Nouvelle édition par G.-H. Bousquet et G. Busino, Librairie Droz, Geneva, pp. 299–345.Google Scholar
18. Pickands, J. (1975). Statistical inference using extreme order statistics. Annals of Statistics, 3: 119–131.Google Scholar
19. R-forge Distributions Core Team (2008). A guide on probability distributions. http://dutangc.free.fr/pub/prob/probdistr-main.pdf Google Scholar
20. Ramsay, C.M. (2006). The distribution of sums of certain I.I.D. Pareto variates. Communications in Statistics—Theory and Methods, 35: 395–405.CrossRefGoogle Scholar
21. Ramsay, C.M. (2007). Exact waiting time and queue size distributions for equilibrium M/G/1 queues with Pareto service. Queueing Systems, 57: 147–155.CrossRefGoogle Scholar
22. Ramsay, C.M. (2008). The distribution of sums of I.I.D. Pareto random variables with arbitrary shape parameter. Communications in Statistics—Theory and Methods, 37: 2177–2184.Google Scholar
23. Ramsay, C.M. (2009). The distribution of compound sums of Pareto distributed losses. Scandinavian Actuarial Journal: 27–37.Google Scholar
24. Wendel, J.G. (1961). The non-absolute convergence of Gil-Pelaez' inversion integral. Annals of Mathematical Statistics, 32: 338–339.Google Scholar
25. Zaliapin, I.V., Kagan, Y.Y., & Schoenberg, F.P. (2005). Approximating the distribution of Pareto sums. Pure and Applied Geophysics, 162: 1187–1228.CrossRefGoogle Scholar