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Conditional expectations given the sum of independent random variables with regularly varying densities

Author

Listed:
  • Denuit, Michel

    (Université catholique de Louvain, LIDAM/ISBA, Belgium)

  • Ortega-Jimenez, Patricia

    (Université catholique de Louvain, LIDAM/ISBA, Belgium)

  • Robert, Christian Y.

Abstract

Stochastic monotonicity of two independent random variables X and Y given the value of their sum S = X + Y has been linked to log-concave densities since Efron (1965). However, the log-concavity assumption is not realistic in some applications because it excludes heavy- tailed distributions. This paper considers random variables with regularly varying densities to illustrate how heavy tails can lead to a non-monotonic behavior for the conditional expectation mX(s) = E[X|S = s], which turns out to be problematic in risk sharing or signal processing (including industry loss warranties or parametric insurance, for instance). This paper first aims to identify situations where a non-monotonic behavior appears according to the tail-heaviness of X and Y . Secondly the paper aims to study the asymptotic behavior of mX (s) as the value s of the sum gets large. The analysis is then extended to zero-augmented probability distributions, commonly encountered in applications to insurance and to sums of more than two random variables. Consequences for signal processing and risk sharing are discussed. Many numerical examples illustrate the results.

Suggested Citation

  • Denuit, Michel & Ortega-Jimenez, Patricia & Robert, Christian Y., 2024. "Conditional expectations given the sum of independent random variables with regularly varying densities," LIDAM Discussion Papers ISBA 2024006, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    • Handle: RePEc:aiz:louvad:2024006
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    References listed on IDEAS

    as
    1. Søren Asmussen & Jaakko Lehtomaa, 2017. "Distinguishing Log-Concavity from Heavy Tails," Risks, MDPI, vol. 5(1), pages 1-14, February.
    2. Denuit, Michel & Robert, Christian Y., 2020. "Large-Loss Behavior of Conditional Mean Risk Sharing," LIDAM Reprints ISBA 2020021, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    3. Denuit, Michel, 2019. "Size-Biased Transform And Conditional Mean Risk Sharing, With Application To P2p Insurance And Tontines," ASTIN Bulletin, Cambridge University Press, vol. 49(3), pages 591-617, September.
    4. Denuit, Michel & Dhaene, Jan, 2012. "Convex order and comonotonic conditional mean risk sharing," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 265-270.
    5. Nadine Gatzert & Sebastian Pokutta & Nikolai Vogl, 2019. "Convergence Of Capital And Insurance Markets: Consistent Pricing Of Index‐Linked Catastrophe Loss Instruments," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 86(1), pages 39-72, March.
    6. Denuit, Michel, 2019. "Size-biased transform and conditional mean risk sharing, with application to P2P insurance and tontines," LIDAM Discussion Papers ISBA 2019010, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    7. Denuit, Michel, 2019. "Size-biased transform and conditional mean risk sharing, with application to P2P insurance and tontines," LIDAM Reprints ISBA 2019038, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    8. Denuit, Michel & Robert, Christian Y., 2020. "Large-Loss Behavior Of Conditional Mean Risk Sharing," ASTIN Bulletin, Cambridge University Press, vol. 50(3), pages 1093-1122, September.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Log-concavity ; asymptotic smoothness ; size-bias transform ; noisy signal ; risk sharing ; zero-augmented distributions;
    All these keywords.

    JEL classification:

    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

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