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Precise large deviations of aggregate claims with arbitrary dependence between claim sizes and waiting times

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  • Chen, Yiqing
  • White, Toby
  • Yuen, Kam Chuen

Abstract

Consider a renewal risk model in which claim sizes and interarrival times correspondingly form a sequence of independent, identically distributed, and nonnegative random pairs with a generic pair (X,θ). Chen and Yuen (2012) studied precise large deviations of aggregate claims in this model under the assumption that (X,θ) obeys a dependence structure described via a stochastic boundedness condition on the waiting time θ for a large claim X. That assumption unfortunately leads to asymptotic independence between X and θ and hence considerably limits the usefulness of the result obtained there. In this short paper, we make an effort to avoid that assumption by allowing X and θ to be arbitrarily dependent. As by-products, we propose two novel applications of the main result, one to pricing insurance futures and the other to approximating both the value at risk and expected shortfall of aggregate claims.

Suggested Citation

  • Chen, Yiqing & White, Toby & Yuen, Kam Chuen, 2021. "Precise large deviations of aggregate claims with arbitrary dependence between claim sizes and waiting times," Insurance: Mathematics and Economics, Elsevier, vol. 97(C), pages 1-6.
    • Handle: RePEc:eee:insuma:v:97:y:2021:i:c:p:1-6
    DOI: 10.1016/j.insmatheco.2020.12.003
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    References listed on IDEAS

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    1. Shen, Xinmei & Xu, Menghao & Mills, Ebenezer Fiifi Emire Atta, 2016. "Precise large deviation results for sums of sub-exponential claims in a size-dependent renewal risk model," Statistics & Probability Letters, Elsevier, vol. 114(C), pages 6-13.
    2. Chen, Yiqing & Yuen, Kam C., 2012. "Precise large deviations of aggregate claims in a size-dependent renewal risk model," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 457-461.
    3. Kaas, Rob & Tang, Qihe, 2005. "A large deviation result for aggregate claims with dependent claim occurrences," Insurance: Mathematics and Economics, Elsevier, vol. 36(3), pages 251-259, June.
    4. Alexander J. McNeil & Rüdiger Frey & Paul Embrechts, 2015. "Quantitative Risk Management: Concepts, Techniques and Tools Revised edition," Economics Books, Princeton University Press, edition 2, number 10496.
    5. Zhu, Wenge, 2017. "Wanting robustness in insurance: A model of catastrophe risk pricing and its empirical test," Insurance: Mathematics and Economics, Elsevier, vol. 77(C), pages 14-23.
    6. Cossette, Hélène & Marceau, Etienne & Marri, Fouad, 2008. "On the compound Poisson risk model with dependence based on a generalized Farlie-Gumbel-Morgenstern copula," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 444-455, December.
    7. Asimit, Alexandru V. & Li, Jinzhu, 2018. "Systemic Risk: An Asymptotic Evaluation," ASTIN Bulletin, Cambridge University Press, vol. 48(2), pages 673-698, May.
    8. Fu, Ke-Ang & Ng, Cheuk Yin Andrew, 2014. "Asymptotics for the ruin probability of a time-dependent renewal risk model with geometric Lévy process investment returns and dominatedly-varying-tailed claims," Insurance: Mathematics and Economics, Elsevier, vol. 56(C), pages 80-87.
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    Cited by:

    1. Gao, Qingwu & Lin, Jia’nan & Liu, Xijun, 2023. "Large deviations of aggregate amount of claims in compound risk model with arbitrary dependence between claim sizes and waiting times," Statistics & Probability Letters, Elsevier, vol. 197(C).
    2. Yuan, Meng & Lu, Dawei, 2022. "Precise large deviation for sums of sub-exponential claims with the m-dependent semi-Markov type structure," Statistics & Probability Letters, Elsevier, vol. 185(C).
    3. Fu, Ke-Ang & Liu, Yang & Wang, Jiangfeng, 2022. "Precise large deviations in a bidimensional risk model with arbitrary dependence between claim-size vectors and waiting times," Statistics & Probability Letters, Elsevier, vol. 184(C).
    4. Jinyu Zhou & Jigao Yan & Dongya Cheng, 2024. "Strong consistency of tail value-at-risk estimator and corresponding general results under widely orthant dependent samples," Statistical Papers, Springer, vol. 65(6), pages 3357-3394, August.

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