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Distinguishing Log-Concavity from Heavy Tails

Author

Listed:
  • Søren Asmussen

    (Department of Mathematics, Aarhus University, Ny Munkegade 118, DK-8000 Aarhus C, Denmark)

  • Jaakko Lehtomaa

    (Department of Mathematics, Aarhus University, Ny Munkegade 118, DK-8000 Aarhus C, Denmark)

Abstract

Well-behaved densities are typically log-convex with heavy tails and log-concave with light ones. We discuss a benchmark for distinguishing between the two cases, based on the observation that large values of a sum X 1 + X 2 occur as result of a single big jump with heavy tails whereas X 1 , X 2 are of equal order of magnitude in the light-tailed case. The method is based on the ratio | X 1 − X 2 | / ( X 1 + X 2 ) , for which sharp asymptotic results are presented as well as a visual tool for distinguishing between the two cases. The study supplements modern non-parametric density estimation methods where log-concavity plays a main role, as well as heavy-tailed diagnostics such as the mean excess plot.

Suggested Citation

  • Søren Asmussen & Jaakko Lehtomaa, 2017. "Distinguishing Log-Concavity from Heavy Tails," Risks, MDPI, vol. 5(1), pages 1-14, February.
    • Handle: RePEc:gam:jrisks:v:5:y:2017:i:1:p:10-:d:89508
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    References listed on IDEAS

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    Cited by:

    1. Miriam Hägele & Jaakko Lehtomaa, 2021. "Large Deviations for a Class of Multivariate Heavy-Tailed Risk Processes Used in Insurance and Finance," JRFM, MDPI, vol. 14(5), pages 1-18, May.
    2. 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).
    3. Miriam Hägele & Jaakko Lehtomaa, 2023. "On the Identification of the Riskiest Directional Components from Multivariate Heavy-Tailed Data," Risks, MDPI, vol. 11(7), pages 1-18, July.

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