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Convergence and asymptotic variance of bootstrapped finite-time ruin probabilities with partly shifted risk processes

Author

Listed:
  • Stéphane Loisel

    (SAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon)

  • Christian Mazza

    (Département de Mathématiques - Albert-Ludwigs-Universität Freiburg)

  • Didier Rullière

    (SAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon)

Abstract

The classical risk model is considered and a sensitivity analysis of finite-time ruin probabilities is carried out. We prove the weak convergence of a sequence of empirical finite-time ruin probabilities. So-called partly shifted risk processes are introduced, and used to derive an explicit expression of the asymptotic variance of the considered estimator. This provides a clear representation of the influence function associated with finite time ruin probabilities, giving a useful tool to quantify estimation risk according to new regulations.

Suggested Citation

  • Stéphane Loisel & Christian Mazza & Didier Rullière, 2009. "Convergence and asymptotic variance of bootstrapped finite-time ruin probabilities with partly shifted risk processes," Post-Print hal-00168716, HAL.
  • Handle: RePEc:hal:journl:hal-00168716
    DOI: 10.1016/j.insmatheco.2009.08.003
    Note: View the original document on HAL open archive server: https://hal.science/hal-00168716
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    References listed on IDEAS

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    1. Loisel, Stéphane & Mazza, Christian & Rullière, Didier, 2008. "Robustness analysis and convergence of empirical finite-time ruin probabilities and estimation risk solvency margin," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 746-762, April.
    2. H. Panjer, Harry & Shaun Wang,, 1993. "On the Stability of Recursive Formulas," ASTIN Bulletin, Cambridge University Press, vol. 23(2), pages 227-258, November.
    3. Mazza, Christian & Rulliere, Didier, 2004. "A link between wave governed random motions and ruin processes," Insurance: Mathematics and Economics, Elsevier, vol. 35(2), pages 205-222, October.
    4. Biard, Romain & Lefèvre, Claude & Loisel, Stéphane, 2008. "Impact of correlation crises in risk theory: Asymptotics of finite-time ruin probabilities for heavy-tailed claim amounts when some independence and stationarity assumptions are relaxed," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 412-421, December.
    5. Stéphane Loisel & Claude Lefèvre, 2009. "Finite-Time Ruin Probabilities for Discrete, Possibly Dependent, Claim Severities," Post-Print hal-00201377, HAL.
    6. Claude Lefèvre & Stéphane Loisel, 2009. "Finite-Time Ruin Probabilities for Discrete, Possibly Dependent, Claim Severities," Methodology and Computing in Applied Probability, Springer, vol. 11(3), pages 425-441, September.
    7. Picard, Philippe & Lefevre, Claude, 1998. "The moments of ruin time in the classical risk model with discrete claim size distribution," Insurance: Mathematics and Economics, Elsevier, vol. 23(2), pages 157-172, November.
    8. Marceau, Etienne & Rioux, Jacques, 2001. "On robustness in risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 29(2), pages 167-185, October.
    9. Claude Lefèvre & Stéphane Loisel, 2008. "On Finite-Time Ruin Probabilities for Classical Risk Models," Post-Print hal-00168958, HAL.
    10. Romain Biard & Claude Lefèvre & Stéphane Loisel, 2008. "Impact of correlation crises in risk theory," Post-Print hal-00308782, HAL.
    11. Frees, Edward W., 1986. "Nonparametric Estimation of the Probability of Ruin," ASTIN Bulletin, Cambridge University Press, vol. 16(S1), pages 81-90, April.
    12. Stéphane Loisel & Nicolas Privault, 2009. "Sensitivity analysis and density estimation for finite-time ruin probabilities," Post-Print hal-00201347, HAL.
    13. Ignatov, Zvetan G. & Kaishev, Vladimir K. & Krachunov, Rossen S., 2001. "An improved finite-time ruin probability formula and its Mathematica implementation," Insurance: Mathematics and Economics, Elsevier, vol. 29(3), pages 375-386, December.
    14. Croux, Kristof & Veraverbeke, Noel, 1990. "Nonparametric estimators for the probability of ruin," Insurance: Mathematics and Economics, Elsevier, vol. 9(2-3), pages 127-130, September.
    15. Hipp, Christian, 1989. "Estimators and Bootstrap Confidence Intervals for Ruin Probabilities," ASTIN Bulletin, Cambridge University Press, vol. 19(1), pages 57-70, April.
    16. Panjer, Harry H., 1981. "Recursive Evaluation of a Family of Compound Distributions," ASTIN Bulletin, Cambridge University Press, vol. 12(1), pages 22-26, June.
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    Cited by:

    1. Stéphane Loisel & Nicolas Privault, 2009. "Sensitivity analysis and density estimation for finite-time ruin probabilities," Post-Print hal-00201347, HAL.
    2. Fabrice Borel-Mathurin & Nicole El Karoui & Stéphane Loisel & Julien Vedani, 2020. "Locality in time of the European insurance regulation "risk-neutral" valuation framework, a pre-and post-Covid analysis and further developments," Working Papers hal-02905181, HAL.
    3. Li Qin & Susan M. Pitts, 2012. "Nonparametric Estimation of the Finite-Time Survival Probability with Zero Initial Capital in the Classical Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 14(4), pages 919-936, December.
    4. Romain Biard & Stéphane Loisel & Claudio Macci & Noel Veraverbeke, 2010. "Asymptotic behavior of the finite-time expected time-integrated negative part of some risk processes and optimal reserve allocation," Post-Print hal-00372525, HAL.

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