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A parametric alternative to the Hill estimator for heavy-tailed distributions

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  • Kim, Joseph H.T.
  • Kim, Joocheol

Abstract

Despite its wide use, the Hill estimator and its plot remain to be difficult to use in Extreme Value Theory (EVT) due to substantial sampling variations in extreme sample quantiles. In this paper, we propose a new plot we call the eigenvalue plot which can be seen as a generalization of the Hill plot. The theory behind the plot is based on a heavy-tailed parametric distribution class called the scaled Log phase-type (LogPH) distributions, a generalization of the ordinary LogPH distribution class which was previously used to model insurance claims data. We show that its tail property and moment condition are well aligned with EVT. Based on our findings, we construct the eigenvalue plot from fitting a shifted PH distribution to the excess log data with a minimal phase size. Through various numerical examples we illustrate and compare our method against the Hill plot.

Suggested Citation

  • Kim, Joseph H.T. & Kim, Joocheol, 2015. "A parametric alternative to the Hill estimator for heavy-tailed distributions," Journal of Banking & Finance, Elsevier, vol. 54(C), pages 60-71.
    • Handle: RePEc:eee:jbfina:v:54:y:2015:i:c:p:60-71
    DOI: 10.1016/j.jbankfin.2014.12.020
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    References listed on IDEAS

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    1. Einmahl, J. H.J. & Dekkers, A. L.M. & de Haan, L., 1989. "A moment estimator for the index of an extreme-value distribution," Other publications TiSEM 81970cb3-5b7a-4cad-9bf6-2, Tilburg University, School of Economics and Management.
    2. Ahn, Soohan & Kim, Joseph H.T. & Ramaswami, Vaidyanathan, 2012. "A new class of models for heavy tailed distributions in finance and insurance risk," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 43-52.
    3. Hall, Peter, 1990. "Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems," Journal of Multivariate Analysis, Elsevier, vol. 32(2), pages 177-203, February.
    4. Bladt, Mogens, 2005. "A Review on Phase-type Distributions and their Use in Risk Theory," ASTIN Bulletin, Cambridge University Press, vol. 35(1), pages 145-161, May.
    5. 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.
    6. Ibragimov, Marat & Ibragimov, Rustam & Kattuman, Paul, 2013. "Emerging markets and heavy tails," Journal of Banking & Finance, Elsevier, vol. 37(7), pages 2546-2559.
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    Cited by:

    1. Lehtomaa, Jaakko & Resnick, Sidney I., 2020. "Asymptotic independence and support detection techniques for heavy-tailed multivariate data," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 262-277.

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    More about this item

    Keywords

    Hill estimator; Tail index; Extreme value theory; Generalized Pareto distribution; Scaled Log phase-type distribution;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • F31 - International Economics - - International Finance - - - Foreign Exchange

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