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A New Parameter Estimator for the Generalized Pareto Distribution under the Peaks over Threshold Framework

Author

Listed:
  • Xu Zhao

    (College of Applied Sciences, Beijing University of Technology, Beijing 100124, China)

  • Zhongxian Zhang

    (College of Applied Sciences, Beijing University of Technology, Beijing 100124, China)

  • Weihu Cheng

    (College of Applied Sciences, Beijing University of Technology, Beijing 100124, China)

  • Pengyue Zhang

    (Department of Biomedical Informatics, College of Medicine, The Ohio State University, Columbus, OH 43210, USA)

Abstract

Techniques used to analyze exceedances over a high threshold are in great demand for research in economics, environmental science, and other fields. The generalized Pareto distribution (GPD) has been widely used to fit observations exceeding the tail threshold in the peaks over threshold (POT) framework. Parameter estimation and threshold selection are two critical issues for threshold-based GPD inference. In this work, we propose a new GPD-based estimation approach by combining the method of moments and likelihood moment techniques based on the least squares concept, in which the shape and scale parameters of the GPD can be simultaneously estimated. To analyze extreme data, the proposed approach estimates the parameters by minimizing the sum of squared deviations between the theoretical GPD function and its expectation. Additionally, we introduce a recently developed stopping rule to choose the suitable threshold above which the GPD asymptotically fits the exceedances. Simulation studies show that the proposed approach performs better or similar to existing approaches, in terms of bias and the mean square error, in estimating the shape parameter. In addition, the performance of three threshold selection procedures is assessed by estimating the value-at-risk (VaR) of the GPD. Finally, we illustrate the utilization of the proposed method by analyzing air pollution data. In this analysis, we also provide a detailed guide regarding threshold selection.

Suggested Citation

  • Xu Zhao & Zhongxian Zhang & Weihu Cheng & Pengyue Zhang, 2019. "A New Parameter Estimator for the Generalized Pareto Distribution under the Peaks over Threshold Framework," Mathematics, MDPI, vol. 7(5), pages 1-18, May.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:5:p:406-:d:228962
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    References listed on IDEAS

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    1. Castillo, Joan del & Serra, Isabel, 2015. "Likelihood inference for generalized Pareto distribution," Computational Statistics & Data Analysis, Elsevier, vol. 83(C), pages 116-128.
    2. Abdelaati Daouia & Stéphane Girard & Gilles Stupfler, 2018. "Estimation of tail risk based on extreme expectiles," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 80(2), pages 263-292, March.
    3. P. Zea Bermudez & M. Turkman, 2003. "Bayesian approach to parameter estimation of the generalized pareto distribution," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 12(1), pages 259-277, June.
    4. Max Grazier G'Sell & Stefan Wager & Alexandra Chouldechova & Robert Tibshirani, 2016. "Sequential selection procedures and false discovery rate control," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(2), pages 423-444, March.
    5. Song, Jongwoo & Song, Seongjoo, 2012. "A quantile estimation for massive data with generalized Pareto distribution," Computational Statistics & Data Analysis, Elsevier, vol. 56(1), pages 143-150, January.
    6. Pavel V. Shevchenko, 2010. "Implementing loss distribution approach for operational risk," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 26(3), pages 277-307, May.
    7. Park, Myung Hyun & Kim, Joseph H.T., 2016. "Estimating extreme tail risk measures with generalized Pareto distribution," Computational Statistics & Data Analysis, Elsevier, vol. 98(C), pages 91-104.
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