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Joint Asymptotic Distributions of Smallest and Largest Insurance Claims

Author

Listed:
  • Hansjörg Albrecher

    (Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, 1015 Lausanne, Switzerland
    Swiss Finance Institute, Lausanne 1015, Switzerland)

  • Christian Y. Robert

    (Université de Lyon, Université Lyon 1, Institut de Science Financière et d'Assurances,Lyon 69007, France)

  • Jef L. Teugels

    (Department of Mathematics, University of Leuven, Leuven 3001, Belgium)

Abstract

Assume that claims in a portfolio of insurance contracts are described by independent and identically distributed random variables with regularly varying tails and occur according to a near mixed Poisson process. We provide a collection of results pertaining to the joint asymptotic Laplace transforms of the normalised sums of the smallest and largest claims, when the length of the considered time interval tends to infinity. The results crucially depend on the value of the tail index of the claim distribution, as well as on the number of largest claims under consideration.

Suggested Citation

  • Hansjörg Albrecher & Christian Y. Robert & Jef L. Teugels, 2014. "Joint Asymptotic Distributions of Smallest and Largest Insurance Claims," Risks, MDPI, vol. 2(3), pages 1-26, July.
    • Handle: RePEc:gam:jrisks:v:2:y:2014:i:3:p:289-314:d:38776
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    References listed on IDEAS

    as
    1. Sophie A. Ladoucette & Jef L. Teugels, 2007. "Asymptotics for Ratios with Applications to Reinsurance," Methodology and Computing in Applied Probability, Springer, vol. 9(2), pages 225-242, June.
    2. Ladoucette, Sophie A., 2007. "Asymptotic behavior of the moments of the ratio of the random sum of squares to the square of the random sum," Statistics & Probability Letters, Elsevier, vol. 77(10), pages 1021-1033, June.
    3. Ammeter, Hans, 1964. "Note Concerning the Distribution Function of the Total Loss Excluding the Largest Individual Claims," ASTIN Bulletin, Cambridge University Press, vol. 3(2), pages 132-143, August.
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    Cited by:

    1. Asimit, Alexandru V. & Chen, Yiqing, 2015. "Asymptotic results for conditional measures of association of a random sum," Insurance: Mathematics and Economics, Elsevier, vol. 60(C), pages 11-18.
    2. Søren Asmussen & Jaakko Lehtomaa, 2017. "Distinguishing Log-Concavity from Heavy Tails," Risks, MDPI, vol. 5(1), pages 1-14, February.
    3. Albrecher, Hansjörg & García Flores, Brandon, 2022. "Asymptotic analysis of generalized Greenwood statistics for very heavy tails," Statistics & Probability Letters, Elsevier, vol. 185(C).

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