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Asymptotically Normal Estimators of the Ruin Probability for Lévy Insurance Surplus from Discrete Samples

Author

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  • Yasutaka Shimizu

    (Department of Applied Mathematics, Waseda University, Shinjuku City, Tokyo 169-8555, Japan)

  • Zhimin Zhang

    (College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China)

Abstract

A statistical inference for ruin probability from a certain discrete sample of the surplus is discussed under a spectrally negative Lévy insurance risk. We consider the Laguerre series expansion of ruin probability, and provide an estimator for any of its partial sums by computing the coefficients of the expansion. We show that the proposed estimator is asymptotically normal and consistent with the optimal rate of convergence and estimable asymptotic variance. This estimator enables not only a point estimation of ruin probability but also an approximated interval estimation and testing hypothesis.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jrisks:v:7:y:2019:i:2:p:37-:d:219722
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    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. Shimizu, Yasutaka & Tanaka, Shuji, 2018. "Dynamic risk measures for stochastic asset processes from ruin theory," Annals of Actuarial Science, Cambridge University Press, vol. 12(2), pages 249-268, September.
    4. Julien Trufin & Hansjoerg Albrecher & Michel M Denuit, 2011. "Properties of a Risk Measure Derived from Ruin Theory," The Geneva Risk and Insurance Review, Palgrave Macmillan;International Association for the Study of Insurance Economics (The Geneva Association), vol. 36(2), pages 174-188, December.
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    Citations

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    Cited by:

    1. Aili Zhang & Ping Chen & Shuanming Li & Wenyuan Wang, 2020. "Risk Modelling on Liquidations with L\'{e}vy Processes," Papers 2007.01426, arXiv.org.
    2. Su, Wen & Yong, Yaodi, 2024. "Estimating a VaR-type ruin measure by Laguerre series expansion in classical compound Poisson risk model," Statistics & Probability Letters, Elsevier, vol. 205(C).
    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.
    4. Xie, Jiayi & Cui, Zhenyu & Zhang, Zhimin, 2022. "Some new infinite series expansions for the first passage time densities in a jump diffusion model with phase-type jumps," Applied Mathematics and Computation, Elsevier, vol. 429(C).
    5. He, Yue & Kawai, Reiichiro & Shimizu, Yasutaka & Yamazaki, Kazutoshi, 2023. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Insurance: Mathematics and Economics, Elsevier, vol. 109(C), pages 1-28.
    6. Zhang, Aili & Chen, Ping & Li, Shuanming & Wang, Wenyuan, 2022. "Risk modelling on liquidations with Lévy processes," Applied Mathematics and Computation, Elsevier, vol. 412(C).
    7. Andrius Grigutis & Jonas Šiaulys, 2020. "Ultimate Time Survival Probability in Three-Risk Discrete Time Risk Model," Mathematics, MDPI, vol. 8(2), pages 1-30, January.
    8. 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.

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