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Nonparametric estimation for the ruin probability in a Lévy risk model under low-frequency observation

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  • Zhang, Zhimin
  • Yang, Hailiang

Abstract

In this paper, we propose a nonparametric estimator for the ruin probability in a spectrally negative Lévy risk model based on low-frequency observation. The estimator is constructed via the Fourier transform of the ruin probability. The convergence rates of the estimator are studied for large sample size. Some simulation results are also given to show the performance of the proposed method when the sample size is finite.

Suggested Citation

  • Zhang, Zhimin & Yang, Hailiang, 2014. "Nonparametric estimation for the ruin probability in a Lévy risk model under low-frequency observation," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 168-177.
    • Handle: RePEc:eee:insuma:v:59:y:2014:i:c:p:168-177
    DOI: 10.1016/j.insmatheco.2014.09.006
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    1. Kappus, Johanna & Reiß, Markus, 2010. "Estimation of the characteristics of a Lévy process observed at arbitrary frequency," SFB 649 Discussion Papers 2010-015, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    2. Kappus, Johanna, 2014. "Adaptive nonparametric estimation for Lévy processes observed at low frequency," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 730-758.
    3. S. Pitts, 1994. "Nonparametric estimation of compound distributions with applications in insurance," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 46(3), pages 537-555, September.
    4. Zhang, Zhimin & Yang, Hailiang, 2013. "Nonparametric estimate of the ruin probability in a pure-jump Lévy risk model," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 24-35.
    5. Frees, Edward W., 1986. "Nonparametric Estimation of the Probability of Ruin," ASTIN Bulletin, Cambridge University Press, vol. 16(S1), pages 81-90, April.
    6. Shimizu, Yasutaka, 2009. "A new aspect of a risk process and its statistical inference," Insurance: Mathematics and Economics, Elsevier, vol. 44(1), pages 70-77, February.
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    8. Johanna Kappus & Markus Reiß, 2010. "Estimation of the characteristics of a Lévy process observed at arbitrary frequency," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 64(s1), pages 314-328.
    9. 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.
    10. Hipp, Christian, 1989. "Estimators and Bootstrap Confidence Intervals for Ruin Probabilities," ASTIN Bulletin, Cambridge University Press, vol. 19(1), pages 57-70, April.
    11. Comte, F. & Genon-Catalot, V., 2009. "Nonparametric estimation for pure jump Lévy processes based on high frequency data," Stochastic Processes and their Applications, Elsevier, vol. 119(12), pages 4088-4123, December.
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    Cited by:

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    2. Kang Hu & Ya Huang & Yingchun Deng, 2023. "Estimating the Gerber–Shiu Function in the Two-Sided Jumps Risk Model by Laguerre Series Expansion," Mathematics, MDPI, vol. 11(9), pages 1-30, April.
    3. Yue He & Reiichiro Kawai & Yasutaka Shimizu & Kazutoshi Yamazaki, 2022. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Papers 2203.10680, arXiv.org, revised Dec 2022.
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    5. Yunyun Wang & Wenguang Yu & Yujuan Huang & Xinliang Yu & Hongli Fan, 2019. "Estimating the Expected Discounted Penalty Function in a Compound Poisson Insurance Risk Model with Mixed Premium Income," Mathematics, MDPI, vol. 7(3), pages 1-25, March.
    6. Yuan Gao & Honglong You, 2021. "The Speed of Convergence of the Threshold Estimator of Ruin Probability under the Tempered α -Stable Lévy Subordinator," Mathematics, MDPI, vol. 9(21), pages 1-9, October.
    7. Wen Su & Wenguang Yu, 2020. "Asymptotically Normal Estimators of the Gerber-Shiu Function in Classical Insurance Risk Model," Mathematics, MDPI, vol. 8(10), pages 1-11, September.
    8. Shimizu, Yasutaka & Zhang, Zhimin, 2017. "Estimating Gerber–Shiu functions from discretely observed Lévy driven surplus," Insurance: Mathematics and Economics, Elsevier, vol. 74(C), pages 84-98.
    9. Yang, Yang & Su, Wen & Zhang, Zhimin, 2019. "Estimating the discounted density of the deficit at ruin by Fourier cosine series expansion," Statistics & Probability Letters, Elsevier, vol. 146(C), pages 147-155.
    10. Wen Su & Yunyun Wang, 2021. "Estimating the Gerber-Shiu Function in Lévy Insurance Risk Model by Fourier-Cosine Series Expansion," Mathematics, MDPI, vol. 9(12), pages 1-18, June.

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