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

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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.

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References

  • Broeckx, F., Goovaerts, M.J., De Vylder, F.: Ordering of risks and ruin probabilities. Insur. Mathe. Econ. 5, 35–40 (1986)

    Article  MATH  Google Scholar 

  • Cai, J., Wu, Y.: Some improvements on the Lundberg’s bound for the ruin probability. Stat. Probab. Lett. 33, 395–403 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • De Vylder, F., Goovaerts, M.J.: Bounds for classical ruin probabilities. Insur. Math. Econ. 3, 121–131 (1984)

    Article  MATH  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • Feller, W.: An Introduction to Probability Theory and its Applications, vol. II. Wiley, New York (1971)

    MATH  Google Scholar 

  • 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)

    MATH  MathSciNet  Google Scholar 

  • Gerber, H.: Martingales in risk theory. Ver. Schweiz. Versicher. Mathematiker Mitt. 73, 205–216 (1973)

    MathSciNet  Google Scholar 

  • Grandell, J.: Aspects of Risk Theory. Springer, New York (1991)

    MATH  Google Scholar 

  • Lin, X.: Tail of compound distributions and excess time. J. Appl. Probab. 33, 184–195 (1997)

    Article  Google Scholar 

  • Lundberg, F.: Approximerad Framställning av Sannolikhetsfunktionen, II. Almqvist & Wiksell, Uppsala (1903)

    Google Scholar 

  • Pitts, S., Grübel, R., Embrechts, P.: Confidence bound for the adjustment coefficient. Adv. Appl. Probab. 28, 802–827 (1996)

    Article  MATH  Google Scholar 

  • Ross, S.: Bounds on the delay distribution in GI/G/1 queues. J. Appl. Probab. 11, 417–421 (1974)

    Article  MATH  Google Scholar 

  • Serfling, R.J.: Approximation Theorems of Mathematical Statistics. Wiley, New York (1980)

    Book  MATH  Google Scholar 

  • Stoyan, D.: Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester (1983)

    MATH  Google Scholar 

  • Willmot, G.E.: Refinements and distributional generalizations of Lundberg’s inequality. Insur. Math. Econ. 15, 49–63 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  • Willmot, G.E.: A non-exponential generalization of an inequality arising in queueing and insurance risk. J. Appl. Probab. 33, 176–183 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  • Willmot, G.E.: On the relationship between bounds on the tails of compound distributions. Insur. Math. Econ. 19, 95–103 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  • Willmot, G.E., Lin, X.S.: Lundberg bounds on the tails of compound distributions. J. Appl. Probab. 31, 743–756 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  • Willmot, G.E., Lin, X.S.: Simplified bounds on the tails of compound distributions. J. Appl. Probab. 34, 127–133 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  • Willmot, G.E. Lin, X.S.: Lundberg approximations for compound distributions with insurance applications. Springer, Berlin (2000)

    MATH  Google Scholar 

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Correspondence to Pier Luigi Conti.

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Conti, P.L., Masiello, E. Nonparametric statistical analysis of an upper bound of the ruin probability under large claims. Extremes 13, 439–461 (2010). https://doi.org/10.1007/s10687-009-0094-6

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  • DOI: https://doi.org/10.1007/s10687-009-0094-6

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