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A Review of More than One Hundred Pareto-Tail Index Estimators

Author

Listed:
  • Igor Fedotenkov

    (European Commission)

Abstract

Heavy-tailed distributions are often encountered in economics, finance, biology, telecommunications, geology, etc. The heaviness of a tail is measured by a tail index. Numerous methods for tail index estimation have been proposed. This paper reviews more than one hundred Pareto (and equivalent) tail index estimators. It focuses on univariate estimators for non-truncated data. We discuss the basic features of these estimators and provide their analytical expressions. As samples from heavy-tailed distributions are often analysed by researchers from various sciences, the paper provides nontechnical explanations of the methods, so as to be understood by researchers with intermediate skills in statistics. We also discuss the strengths and weaknesses of the estimators, if known. The main focus of the paper is semi-parametric estimators; however, a number of parametric estimators under-represented in previous reviews are also discussed. The paper can be viewed as a catalog or a reference work on Pareto-tail index estimators. A Monte-Carlo comparison of more than 90 estimators is presented.

Suggested Citation

  • Igor Fedotenkov, 2020. "A Review of More than One Hundred Pareto-Tail Index Estimators," Statistica, Department of Statistics, University of Bologna, vol. 80(3), pages 245-299.
    • Handle: RePEc:bot:rivsta:v:80:y:2020:i:3:p:245-299
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    Cited by:

    1. Tjeerd de Vries & Alexis Akira Toda, 2022. "Capital and Labor Income Pareto Exponents Across Time and Space," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 68(4), pages 1058-1078, December.
    2. González-Sánchez, Mariano & Nave Pineda, Juan M., 2023. "Where is the distribution tail threshold? A tale on tail and copulas in financial risk measurement," International Review of Financial Analysis, Elsevier, vol. 86(C).
    3. Gareth W. Peters & Matteo Malavasi & Georgy Sofronov & Pavel V. Shevchenko & Stefan Trück & Jiwook Jang, 2023. "Cyber loss model risk translates to premium mispricing and risk sensitivity," The Geneva Papers on Risk and Insurance - Issues and Practice, Palgrave Macmillan;The Geneva Association, vol. 48(2), pages 372-433, April.
    4. David Anthoff & Richard S. J. Tol, 2022. "Testing the Dismal Theorem," Journal of the Association of Environmental and Resource Economists, University of Chicago Press, vol. 9(5), pages 885-920.
    5. Gareth W. Peters & Matteo Malavasi & Georgy Sofronov & Pavel V. Shevchenko & Stefan Truck & Jiwook Jang, 2022. "Cyber Loss Model Risk Translates to Premium Mispricing and Risk Sensitivity," Papers 2202.10588, arXiv.org, revised Mar 2023.
    6. Ji Hyung Lee & Yuya Sasaki & Alexis Akira Toda & Yulong Wang, 2022. "Capital and Labor Income Pareto Exponents in the United States, 1916-2019," Papers 2206.04257, arXiv.org.
    7. Magdy El-Adll & H. M. Barakat & Amany Aly & Ning Cai, 2022. "Asymptotic Prediction for Future Observations of a Random Sample of Unknown Continuous Distribution," Complexity, Hindawi, vol. 2022, pages 1-15, April.
    8. Sasaki, Yuya & Wang, Yulong, 2024. "On uniform confidence intervals for the tail index and the extreme quantile," Journal of Econometrics, Elsevier, vol. 244(1).
    9. Laura Liu & Yulong Wang, 2025. "Binary Outcome Models with Extreme Covariates: Estimation and Prediction," Papers 2502.16041, arXiv.org.
    10. Arthur Charpentier & Emmanuel Flachaire, 2022. "Pareto models for top incomes and wealth," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 20(1), pages 1-25, March.
    11. Kan Chen & Tuoyuan Cheng, 2022. "Measuring Tail Risks," Papers 2209.07092, arXiv.org, revised Nov 2022.
    12. Paweł D. Domański, 2024. "Energy-Aware Multicriteria Control Performance Assessment," Energies, MDPI, vol. 17(5), pages 1-18, March.
    13. Gadea Rivas, María Dolores & Gonzalo, Jesús & Olmo, José, 2024. "Testing extreme warming and geographical heterogeneity," UC3M Working papers. Economics 45023, Universidad Carlos III de Madrid. Departamento de Economía.
    14. Priscilla Avegliano & Jaime Simão Sichman, 2023. "Equation-Based Versus Agent-Based Models: Why Not Embrace Both for an Efficient Parameter Calibration?," Journal of Artificial Societies and Social Simulation, Journal of Artificial Societies and Social Simulation, vol. 26(4), pages 1-3.
    15. Tuoyuan Cheng & Saikiran Reddy Poreddy & Kan Chen, 2025. "Tail Risk in Weather Derivatives," Commodities, MDPI, vol. 4(2), pages 1-17, June.
    16. Emrah Altun & Hana N. Alqifari & Kadir Söyler, 2025. "Return Level Prediction with a New Mixture Extreme Value Model," Mathematics, MDPI, vol. 13(17), pages 1-24, August.
    17. Ivanilda Cabral & Frederico Caeiro & M. Ivette Gomes, 2022. "On the comparison of several classical estimators of the extreme value index," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 51(1), pages 179-196, January.
    18. Man, Xinyue & Tang, Qihe, 2024. "Tail risk driven by investment losses and exogenous shocks," ASTIN Bulletin, Cambridge University Press, vol. 54(3), pages 712-737, September.
    19. Frank Cowell & Emmanuel Flachaire, 2021. "Inequality Measurement: Methods and Data," Post-Print hal-03589066, HAL.
    20. Bernhard Klar, 2025. "A Pareto Tail Plot Without Moment Restrictions," The American Statistician, Taylor & Francis Journals, vol. 79(2), pages 156-166, April.
    21. Nelson, Kenric P., 2022. "Independent Approximates enable closed-form estimation of heavy-tailed distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 601(C).

    More about this item

    Keywords

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    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C18 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Methodolical Issues: General
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics

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