Advertisement

Extremes

, Volume 13, Issue 4, pp 439–461 | Cite as

Nonparametric statistical analysis of an upper bound of the ruin probability under large claims

  • Pier Luigi ContiEmail author
  • Esterina Masiello
Article

Abstract

In this paper, the classical Poisson risk model is considered. The claims are supposed to be modeled by heavy-tailed distributions, so that the moment generating function does not exist. The attention is focused on the probability of ruin. We first provide a nonparametric estimator of an upper bound of the ruin probability by Willmot and Lin. Then, its asymptotic behavior is studied. Asymptotic confidence intervals are studied, as well as bootstrap confidence intervals. Results for possibly unstable models are also obtained.

Keywords

Poisson risk model Probability of ruin Nonparametric estimation Asymptotics Heavy-tailed distribution 

AMS 2000 Subject Classifications

62G08 62G20 62G32 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Broeckx, F., Goovaerts, M.J., De Vylder, F.: Ordering of risks and ruin probabilities. Insur. Mathe. Econ. 5, 35–40 (1986)zbMATHCrossRefGoogle Scholar
  2. Cai, J., Wu, Y.: Some improvements on the Lundberg’s bound for the ruin probability. Stat. Probab. Lett. 33, 395–403 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  3. Conti, P.L.: A nonparametric sequential test with power 1 for the ruin probability in some risk models. Stat. Probab. Lett. 72, 333–343 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  4. De Vylder, F., Goovaerts, M.J.: Bounds for classical ruin probabilities. Insur. Math. Econ. 3, 121–131 (1984)zbMATHCrossRefGoogle Scholar
  5. Embrechts, P., Veraverbeke, N.: Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insur. Math. Econ. 1, 55–72 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  6. Feller, W.: An Introduction to Probability Theory and its Applications, vol. II. Wiley, New York (1971)zbMATHGoogle Scholar
  7. Gaver, D.P., Jacobs P.A.: Nonparametric estimation of the probability of a long delay in the M/G/1 queue. J. R. Stat. Soc., B 50, 392–402 (1988)zbMATHMathSciNetGoogle Scholar
  8. Gerber, H.: Martingales in risk theory. Ver. Schweiz. Versicher. Mathematiker Mitt. 73, 205–216 (1973)MathSciNetGoogle Scholar
  9. Grandell, J.: Aspects of Risk Theory. Springer, New York (1991)zbMATHGoogle Scholar
  10. Lin, X.: Tail of compound distributions and excess time. J. Appl. Probab. 33, 184–195 (1997)CrossRefGoogle Scholar
  11. Lundberg, F.: Approximerad Framställning av Sannolikhetsfunktionen, II. Almqvist & Wiksell, Uppsala (1903)Google Scholar
  12. Pitts, S., Grübel, R., Embrechts, P.: Confidence bound for the adjustment coefficient. Adv. Appl. Probab. 28, 802–827 (1996)zbMATHCrossRefGoogle Scholar
  13. Ross, S.: Bounds on the delay distribution in GI/G/1 queues. J. Appl. Probab. 11, 417–421 (1974)zbMATHCrossRefGoogle Scholar
  14. Serfling, R.J.: Approximation Theorems of Mathematical Statistics. Wiley, New York (1980)zbMATHCrossRefGoogle Scholar
  15. Stoyan, D.: Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester (1983)zbMATHGoogle Scholar
  16. Willmot, G.E.: Refinements and distributional generalizations of Lundberg’s inequality. Insur. Math. Econ. 15, 49–63 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  17. Willmot, G.E.: A non-exponential generalization of an inequality arising in queueing and insurance risk. J. Appl. Probab. 33, 176–183 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  18. Willmot, G.E.: On the relationship between bounds on the tails of compound distributions. Insur. Math. Econ. 19, 95–103 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  19. Willmot, G.E., Lin, X.S.: Lundberg bounds on the tails of compound distributions. J. Appl. Probab. 31, 743–756 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  20. Willmot, G.E., Lin, X.S.: Simplified bounds on the tails of compound distributions. J. Appl. Probab. 34, 127–133 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  21. Willmot, G.E. Lin, X.S.: Lundberg approximations for compound distributions with insurance applications. Springer, Berlin (2000)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Dipartimento di Statistica, Probabilità e Statistiche ApplicateSapienza - Università di RomaRomaItaly
  2. 2.Institut de Science Financière et d’AssurancesUniversité de Lyon, Université Claude Bernard Lyon 1LyonFrance

Personalised recommendations