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Testing for the shape parameter of generalized extreme value distribution based on the $$L_q$$ -likelihood ratio statistic

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  • Chao Huang
  • Jin-Guan Lin

    ()

  • Yan-Yan Ren

Abstract

This paper studies the applications of extreme value theory on analysis for closing price data of the Dow-Jones industrial index and Danish fire insurance claims data. The generalized extreme value (GEV) distribution is considered in analyzing the real data, and the hypothesis testing problem for the shape parameter of GEV distribution is investigated based on a new test statistic—the $$L_q$$ -likelihood ratio ( $$L_q$$ R) statistic. The $$L_q$$ R statistic can be treated as a generalized form of the classical likelihood ratio (LR) statistic. We show that the asymptotic behavior of proposed statistic is characterized by the degree of distortion $$q$$ . For small and modest sample sizes, the $$L_q$$ R statistic is still available when $$q$$ is properly chosen. By simulation studies, the proposed statistic not only performs the asymptotic properties, but also outperforms the classical LR statistic as the sample sizes are modest or even small. Meanwhile, the test power based on the new statistic is also simulated by Monte Carlo methods. At last, the models are diagnosed by graphical methods. Copyright Springer-Verlag 2013

Suggested Citation

  • 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.
  • Handle: RePEc:spr:metrik:v:76:y:2013:i:5:p:641-671
    DOI: 10.1007/s00184-012-0409-5
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    References listed on IDEAS

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    1. Chao Huang & Jin-Guan Lin & Yan-Yan Ren, 2012. "Statistical Inferences for Generalized Pareto Distribution Based on Interior Penalty Function Algorithm and Bootstrap Methods and Applications in Analyzing Stock Data," Computational Economics, Springer;Society for Computational Economics, vol. 39(2), pages 173-193, February.
    2. Koopman, Siem Jan & Shephard, Neil & Creal, Drew, 2009. "Testing the assumptions behind importance sampling," Journal of Econometrics, Elsevier, vol. 149(1), pages 2-11, April.
    3. Marimoutou, Velayoudoum & Raggad, Bechir & Trabelsi, Abdelwahed, 2009. "Extreme Value Theory and Value at Risk: Application to oil market," Energy Economics, Elsevier, vol. 31(4), pages 519-530, July.
    4. Hamid Mohtadi & Antu Panini Murshid, 2009. "Risk of catastrophic terrorism: an extreme value approach," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 24(4), pages 537-559.
    5. Davide Ferrari & Sandra Paterlini, 2007. "The Maximum Lq-Likelihood Method: an Application to Extreme Quantile Estimation in Finance," Department of Economics 555, University of Modena and Reggio E., Faculty of Economics "Marco Biagi".
    6. Turan G. Bali, 2003. "An Extreme Value Approach to Estimating Volatility and Value at Risk," The Journal of Business, University of Chicago Press, vol. 76(1), pages 83-108, January.
    7. Lin, Jin-Guan & Huang, Chao & Zhuang, Qing-Yun & Zhu, Li-Ping, 2010. "Estimating generalized state density of near-extreme events and its applications in analyzing stock data," Insurance: Mathematics and Economics, Elsevier, vol. 47(1), pages 13-20, August.
    8. Davide Ferrari & Davide La Vecchia, 2012. "On robust estimation via pseudo-additive information," Biometrika, Biometrika Trust, vol. 99(1), pages 238-244.
    9. Xinsheng Liu, 2007. "Likelihood Ratio Test for and Against Nonlinear Inequality Constraints," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 65(1), pages 93-108, February.
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