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Robust and efficient fitting of the generalized Pareto distribution with actuarial applications in view

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  • Brazauskas, Vytaras
  • Kleefeld, Andreas

Abstract

Due to advances in extreme value theory, the generalized Pareto distribution (GPD) emerged as a natural family for modeling exceedances over a high threshold. Its importance in applications (e.g., insurance, finance, economics, engineering and numerous other fields) can hardly be overstated and is widely documented. However, despite the sound theoretical basis and wide applicability, fitting of this distribution in practice is not a trivial exercise. Traditional methods such as maximum likelihood and method-of-moments are undefined in some regions of the parameter space. Alternative approaches exist but they lack either robustness (e.g., probability-weighted moments) or efficiency (e.g., method-of-medians), or present significant numerical problems (e.g., minimum-divergence procedures). In this article, we propose a computationally tractable method for fitting the GPD, which is applicable for all parameter values and offers competitive trade-offs between robustness and efficiency. The method is based on [`]trimmed moments'. Large-sample properties of the new estimators are provided, and their small-sample behavior under several scenarios of data contamination is investigated through simulations. We also study the effect of our methodology on actuarial applications. In particular, using the new approach, we fit the GPD to the Danish insurance data and apply the fitted model to a few risk measurement and ratemaking exercises.

Suggested Citation

  • Brazauskas, Vytaras & Kleefeld, Andreas, 2009. "Robust and efficient fitting of the generalized Pareto distribution with actuarial applications in view," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 424-435, December.
    • Handle: RePEc:eee:insuma:v:45:y:2009:i:3:p:424-435
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    References listed on IDEAS

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    1. Rosario Dell’Aquila & Paul Embrechts, 2006. "Extremes and Robustness: A Contradiction?," Financial Markets and Portfolio Management, Springer;Swiss Society for Financial Market Research, vol. 20(1), pages 103-118, April.
    2. Vytaras Brazauskas, 2009. "Robust and Efficient Fitting of Loss Models," North American Actuarial Journal, Taylor & Francis Journals, vol. 13(3), pages 356-369.
    3. Vytaras Brazauskas & Robert Serfling, 2000. "Robust and Efficient Estimation of the Tail Index of a Single-Parameter Pareto Distribution," North American Actuarial Journal, Taylor & Francis Journals, vol. 4(4), pages 12-27.
    4. McNeil, Alexander J., 1997. "Estimating the Tails of Loss Severity Distributions Using Extreme Value Theory," ASTIN Bulletin, Cambridge University Press, vol. 27(1), pages 117-137, May.
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    Cited by:

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    2. Marc N. Conte & David L. Kelly, 2016. "An Imperfect Storm: Fat-Tailed Hurricane Damages, Insurance and Climate Policy," Working Papers 2016-01, University of Miami, Department of Economics.
    3. Conte, Marc N. & Kelly, David L., 2018. "An imperfect storm: Fat-tailed tropical cyclone damages, insurance, and climate policy," Journal of Environmental Economics and Management, Elsevier, vol. 92(C), pages 677-706.
    4. Su, Jianxi & Furman, Edward, 2017. "Multiple risk factor dependence structures: Distributional properties," Insurance: Mathematics and Economics, Elsevier, vol. 76(C), pages 56-68.
    5. Chao Huang & Jin-Guan Lin & Yan-Yan Ren, 2013. "Testing for the shape parameter of generalized extreme value distribution based on the $$L_q$$ -likelihood ratio statistic," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(5), pages 641-671, July.
    6. Kennickell, Arthur B., 2021. "Chasing the Tail: A Generalized Pareto Distribution Approach to Estimating Wealth Inequality," SocArXiv u3zs2, Center for Open Science.

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