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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Broeckx, F., Goovaerts, M.J., De Vylder, F.: Ordering of risks and ruin probabilities. Insur. Mathe. Econ. 5, 35–40 (1986)
Cai, J., Wu, Y.: Some improvements on the Lundberg’s bound for the ruin probability. Stat. Probab. Lett. 33, 395–403 (1997)
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)
De Vylder, F., Goovaerts, M.J.: Bounds for classical ruin probabilities. Insur. Math. Econ. 3, 121–131 (1984)
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)
Feller, W.: An Introduction to Probability Theory and its Applications, vol. II. Wiley, New York (1971)
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)
Gerber, H.: Martingales in risk theory. Ver. Schweiz. Versicher. Mathematiker Mitt. 73, 205–216 (1973)
Grandell, J.: Aspects of Risk Theory. Springer, New York (1991)
Lin, X.: Tail of compound distributions and excess time. J. Appl. Probab. 33, 184–195 (1997)
Lundberg, F.: Approximerad Framställning av Sannolikhetsfunktionen, II. Almqvist & Wiksell, Uppsala (1903)
Pitts, S., Grübel, R., Embrechts, P.: Confidence bound for the adjustment coefficient. Adv. Appl. Probab. 28, 802–827 (1996)
Ross, S.: Bounds on the delay distribution in GI/G/1 queues. J. Appl. Probab. 11, 417–421 (1974)
Serfling, R.J.: Approximation Theorems of Mathematical Statistics. Wiley, New York (1980)
Stoyan, D.: Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester (1983)
Willmot, G.E.: Refinements and distributional generalizations of Lundberg’s inequality. Insur. Math. Econ. 15, 49–63 (1994)
Willmot, G.E.: A non-exponential generalization of an inequality arising in queueing and insurance risk. J. Appl. Probab. 33, 176–183 (1996)
Willmot, G.E.: On the relationship between bounds on the tails of compound distributions. Insur. Math. Econ. 19, 95–103 (1997)
Willmot, G.E., Lin, X.S.: Lundberg bounds on the tails of compound distributions. J. Appl. Probab. 31, 743–756 (1994)
Willmot, G.E., Lin, X.S.: Simplified bounds on the tails of compound distributions. J. Appl. Probab. 34, 127–133 (1997)
Willmot, G.E. Lin, X.S.: Lundberg approximations for compound distributions with insurance applications. Springer, Berlin (2000)
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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