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Estimation of the Ruin Probability in Infinite Time for Heavy Right-Tailed Losses

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Modern Problems in Insurance Mathematics

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Abstract

The chapter is devoted to the study of asymptotically normal estimators for the ruin probability in infinite time horizon, for insurance models with large initial reserves and heavy-tailed claim distributions. Our considerations are based on the extreme quantile approach. A simulation study illustrates the main results.

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References

  1. Asmussen, S.: Ruin Probabilities. World Scientific, Singapore, New Jersey, London, Hong Kong (2000)

    Google Scholar 

  2. Beirlant, J., Matthys, G., Dierckx, G.: Heavy-tailed distributions and rating. Astin Bull. 31, 37–58 (2001)

    Article  MathSciNet  Google Scholar 

  3. Beirlant, J., Teugels, J.: Asymptotic normality of Hill’s estimator. In: Extreme Value Theory (Oberwolfach, 1987). Lecture Notes in Statistics, vol. 51, Springer, New York, pp 148–155 (1989)

    Google Scholar 

  4. Bingham, N.H., Goldie, C.M., Teugels, J.L., Bingham, N.H., Goldie, C.M.: Regular Variation. Cambridge University Press, Cambridge (1987)

    Book  MATH  Google Scholar 

  5. Cheng, S., Peng, L.: Confidence intervals for the tail index. Bernoulli 7, 751–760 (2001)

    Google Scholar 

  6. Čizěk, P., Härdle, W., Weron, R.: Statistical Tools for Finance and Insurance. Springer, Berlin-Heidelberg (2005)

    MATH  Google Scholar 

  7. Csörgö, M., Csörgö, S., Horveath, L., Mason, D.: Weighted empirical and quantile processes. Ann. Probab. 14, 31–85 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  8. Csörgö, S., Horveath, L., Mason, D.: What portion of the sample makes a partial sum asymptotically stable or normal? Probab. Theory Related Fields 72, 1–16 (1986)

    Article  MATH  Google Scholar 

  9. Csörgö, M., Mason, D.M.: On the asymptotic distribution of weighted uniform empirical and quantile processes in the middle and on the tails. Stochast. Proc. Appl. 21, 119–132 (1985)

    Article  MATH  Google Scholar 

  10. Csörgö, S., Viharos, L.: Estimating the Tail Index. Asymptotic Methods in Probability and Statistics. North-Holland, Amsterdam (1998)

    Google Scholar 

  11. Deheuvels, P., Haeusler, E., Mason, D.: Almost sure convergence of the Hill estimator. Math. Proc. Camb. Philos. Soc. 104, 371–381 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dekkers, A., Einmahl, J., de Haan, L.: A moment estimator for the index of an extreme-value distribution. Ann. Statist. 17, 1833–1855 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dekkers, A., de Haan, L.: Optimal choice of sample fraction in extreme-value estimation. J. Mult. Anal. 47, 173–195 (1993)

    Google Scholar 

  14. Dekkers, A., de Haan, L.: On the estimation of the extreme value index and large quantile estimation. Ann. Statist. 17, 1795–1832 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  15. Dekkers, A., Einmahl, J., de Haan, L.: A moment estimator for the index of an extreme-value distribution. Ann. Statist. 17, 1833–1855 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  16. Drees, H., Kaufmann, E.: Selection of the optimal sample fraction in univariate extreme value estimation. Stochast. Proc. Appl. 75, 149–195 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. de Haan, L., Stadtmüller, U.: Generalized regular variation of second order. J. Australian Math. Soc. 61, 381–395 (1996)

    Article  MATH  Google Scholar 

  18. de Haan, L., Ferreria, A.: Extreme Value Theory: An Introduction. Springer, New York (2006)

    Google Scholar 

  19. Geluk, J., de Haan, L.: Regular Variation, Extension and Tauberian Theorems. CWI Tract 40, Amsterdam (1987)

    Google Scholar 

  20. Hill, B.M.: A simple approach to inference about the tail of a distribution. Ann. Statist. 3, 1136–1174 (1975)

    Article  Google Scholar 

  21. Ihaka, R., Gentleman, R.R.: A language for data analysis and graphics. J. Comput. Graph. Statist. 5, 299–314 (1996)

    Google Scholar 

  22. Mason, D.: Laws of large numbers for sums of extreme values. Ann.Probab 10, 754–764 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  23. Necir, A., Rassoul, A., Zitikis, R.: Estimating the Conditional Tail Expectation in the Case of Heavy-Tailed Losses. JPS, Philadelphia (2010)

    Google Scholar 

  24. Panjer, H., Willmot, G.E.: Insurance Risk Models. Society of actuaries, Michigan (1992)

    Google Scholar 

  25. Peng, L.: Estimating the mean of a heavy-tailed distribution. Stat. Probab. Lett. 52, 255–264 (2001)

    Article  MATH  Google Scholar 

  26. Reiss, R.D., Thomas, M.: Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields, 3rd edn. Birkhäuser Verlag, Basel, Boston, Berlin (2007)

    MATH  Google Scholar 

  27. Rolski, T., Schimidli, H., Schimd, V., Teugels, J.L.: Stochastic Processes for Insurance and Finance. John Wiley and Sons, New York (1999)

    Book  MATH  Google Scholar 

  28. Weissman, I.: Estimation of parameters and large quantiles based on the \(k\) largest observations. J. Amer. Statist. Assoc. 73, 812–815 (1978)

    MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. National High School of Hydraulics, Blida, Algeria

    Abdelaziz Rassoul

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  1. Abdelaziz Rassoul
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Correspondence to Abdelaziz Rassoul .

Editor information

Editors and Affiliations

  1. Department of Mathematics Professor of Mathematical Statistics, Stockholm University, Stockholm, Sweden

    Dmitrii Silvestrov

  2. Department of Mathematics Professor in Mathematical Statistics, Stockholm University, Stockholm, Sweden

    Anders Martin-Löf

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Rassoul, A. (2014). Estimation of the Ruin Probability in Infinite Time for Heavy Right-Tailed Losses. In: Silvestrov, D., Martin-Löf, A. (eds) Modern Problems in Insurance Mathematics. EAA Series. Springer, Cham. https://doi.org/10.1007/978-3-319-06653-0_9

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