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Polynomial Approximations for Bivariate Aggregate Claims Amount Probability Distributions

Author

Listed:
  • Pierre-Olivier Goffard

    (Institut de Mathematique de Marseille, Aix-Marseille University)

  • Stéphane Loisel

    (Université Claude Bernard Lyon 1, Institut de Science Actuarielle et Financière)

  • Denys Pommeret

    (Institut de Mathematique de Marseille, Aix-Marseille University)

Abstract

A numerical method to compute bivariate probability distributions from their Laplace transforms is presented. The method consists in an orthogonal projection of the probability density function with respect to a probability measure that belongs to a Natural Exponential Family with Quadratic Variance Function (NEF-QVF). A particular link to Lancaster probabilities is highlighted. The procedure allows a quick and accurate calculation of probabilities of interest and does not require strong coding skills. Numerical illustrations and comparisons with other methods are provided. This work is motivated by actuarial applications. We aim at recovering the joint distribution of two aggregate claims amounts associated with two insurance policy portfolios that are closely related, and at computing survival functions for reinsurance losses in presence of two non-proportional reinsurance treaties.

Suggested Citation

  • Pierre-Olivier Goffard & Stéphane Loisel & Denys Pommeret, 2017. "Polynomial Approximations for Bivariate Aggregate Claims Amount Probability Distributions," Methodology and Computing in Applied Probability, Springer, vol. 19(1), pages 151-174, March.
    • Handle: RePEc:spr:metcap:v:19:y:2017:i:1:d:10.1007_s11009-015-9470-7
    DOI: 10.1007/s11009-015-9470-7
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    References listed on IDEAS

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    1. Pierre-Olivier Goffard & Stéphane Loisel & Denys Pommeret, 2015. "A polynomial expansion to approximate the ultimate ruin probability in the compound Poisson ruin model," Post-Print hal-00853680, HAL.
    2. Ambagaspitiya, Rohana S., 1999. "On the distributions of two classes of correlated aggregate claims," Insurance: Mathematics and Economics, Elsevier, vol. 24(3), pages 301-308, May.
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    8. Ambagaspitiya, Rohana S., 1998. "On the distribution of a sum of correlated aggregate claims," Insurance: Mathematics and Economics, Elsevier, vol. 23(1), pages 15-19, October.
    9. Vernic, Raluca, 1999. "Recursive Evaluation of Some Bivariate Compound Distributions," ASTIN Bulletin, Cambridge University Press, vol. 29(2), pages 315-325, November.
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    Cited by:

    1. Denys Pommeret & Laurence Reboul, 2019. "Approximating the Probability Density Function of a Transformation of Random Variables," Methodology and Computing in Applied Probability, Springer, vol. 21(2), pages 633-645, June.
    2. Mnatsakanov, Robert M. & Pommeret, Denys, 2024. "On recovering the relative distribution, Part 1: The moment-recovered approach," Statistics & Probability Letters, Elsevier, vol. 208(C).

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