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Measures of kurtosis: inadmissible for asymmetric distributions?

Author

Listed:
  • Andreas Eberl

    (Karlsruhe Institute of Technology (KIT))

  • Bernhard Klar

    (Karlsruhe Institute of Technology (KIT))

Abstract

Skewness and kurtosis are natural characteristics of a distribution. While it has long been recognized that they are more intrinsically entangled than other characteristics like location and dispersion, this has recently been made more explicit by Eberl and Klar (Stat Papers 65:415–433, 2024) with regard to orders of kurtosis. In this paper, we analyze the implications of this entanglement on kurtosis measures in general and for several specific examples. As a key finding, we show that kurtosis measures that are defined in the classical order-based way, which is analogous to measures of location, dispersion and skewness, do not exist. This raises serious doubts about the frequent application of such measures to skewed data. We then consider old and new proposals for kurtosis measures and evaluate under which additional conditions they measure kurtosis in a meaningful way. Some measures also allow more specific representations of the influence of skewness on the measurement of kurtosis than are available in a general setting. This works particularly well for a family of newly introduced density-based kurtosis measures.

Suggested Citation

  • Andreas Eberl & Bernhard Klar, 2025. "Measures of kurtosis: inadmissible for asymmetric distributions?," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 88(3), pages 365-391, April.
    • Handle: RePEc:spr:metrik:v:88:y:2025:i:3:d:10.1007_s00184-024-00959-z
    DOI: 10.1007/s00184-024-00959-z
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    References listed on IDEAS

    as
    1. Eberl, Andreas & Klar, Bernhard, 2020. "Asymptotic distributions and performance of empirical skewness measures," Computational Statistics & Data Analysis, Elsevier, vol. 146(C).
    2. Frank Critchley & M. C. Jones, 2008. "Asymmetry and Gradient Asymmetry Functions: Density‐Based Skewness and Kurtosis," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(3), pages 415-437, September.
    3. Andreas Eberl & Bernhard Klar, 2024. "Centre-free kurtosis orderings for asymmetric distributions," Statistical Papers, Springer, vol. 65(1), pages 415-433, February.
    4. López-Martín, Carmen & Arguedas-Sanz, Raquel & Muela, Sonia Benito, 2022. "A cryptocurrency empirical study focused on evaluating their distribution functions," International Review of Economics & Finance, Elsevier, vol. 79(C), pages 387-407.
    5. Andreas Eberl & Bernhard Klar, 2022. "Expectile‐based measures of skewness," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(1), pages 373-399, March.
    6. Andreas Eberl & Bernhard Klar, 2023. "Stochastic orders and measures of skewness and dispersion based on expectiles," Statistical Papers, Springer, vol. 64(2), pages 509-527, April.
    7. David C. Blest, 2003. "A New Measure of Kurtosis Adjusted for Skewness," Australian & New Zealand Journal of Statistics, Australian Statistical Publishing Association Inc., vol. 45(2), pages 175-179, June.
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