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Nonparametric estimate of the ruin probability in a pure-jump Lévy risk model

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  • Zhang, Zhimin
  • Yang, Hailiang

Abstract

In this paper, we propose a nonparametric estimator of ruin probability in a Lévy risk model. The aggregate claims process X={Xt,≥0} is modeled by a pure-jump Lévy process. Assume that high-frequency observed data on X are available. The estimator is constructed based on the Pollaczek–Khinchin formula and Fourier transform. Risk bounds as well as a data-driven cut-off selection methodology are presented. Simulation studies are also given to show the finite sample performance of our estimator.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:insuma:v:53:y:2013:i:1:p:24-35
    DOI: 10.1016/j.insmatheco.2013.04.004
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    References listed on IDEAS

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    1. 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.
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    Citations

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    Cited by:

    1. Honglong You & Yuan Gao, 2019. "Non-Parametric Threshold Estimation for the Wiener–Poisson Risk Model," Mathematics, MDPI, vol. 7(6), pages 1-11, June.
    2. Chau, K.W. & Yam, S.C.P. & Yang, H., 2015. "Fourier-cosine method for Gerber–Shiu functions," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 170-180.
    3. Chongkai Xie & Honglong You, 2024. "A Threshold Estimator for Ruin Probability Using the Fourier-Cosine Method in the Wiener–Poisson Risk Model," Mathematics, MDPI, vol. 12(18), pages 1-14, September.
    4. Xie, Jiayi & Zhang, Zhimin, 2020. "Statistical estimation for some dividend problems under the compound Poisson risk model," Insurance: Mathematics and Economics, Elsevier, vol. 95(C), pages 101-115.
    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. 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.
    10. 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.
    11. Lee, Wing Yan & Li, Xiaolong & Liu, Fangda & Shi, Yifan & Yam, Sheung Chi Phillip, 2021. "A Fourier-cosine method for finite-time ruin probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 99(C), pages 256-267.
    12. 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.
    13. 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.
    14. Yasutaka Shimizu & Zhimin Zhang, 2019. "Asymptotically Normal Estimators of the Ruin Probability for Lévy Insurance Surplus from Discrete Samples," Risks, MDPI, vol. 7(2), pages 1-22, April.
    15. Oshime, Takayoshi & Shimizu, Yasutaka, 2018. "Parametric inference for ruin probability in the classical risk model," Statistics & Probability Letters, Elsevier, vol. 133(C), pages 28-37.

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