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Flexible multivariate Hill estimators

Author

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  • Dominicy, Yves
  • Heikkilä, Matias
  • Ilmonen, Pauliina
  • Veredas, David

Abstract

Dominicy et al. (2017) introduce a family of Hill estimators for elliptically distributed and heavy tailed random vectors. They propose to use the univariate Hill to a norm of order h of the data. The norms are homogeneous functions of order one. We show that the family of estimators can be generalized to homogeneous functions of any order and, more importantly, that ellipticity is not required. Only multivariate regular variation is needed, as it is preserved under well-behaved homogeneous functions. This enables us to have flexibility in terms of the estimator and the underlying distribution. Consistency and asymptotic normality are shown, and a Monte Carlo study is conducted to assess the finite sample properties under different asymmetric and heavy tailed multivariate distributions. We illustrate the estimators with an application to 10 years of daily data of paid claims from property insurance policies across 15 regions of Belgium.

Suggested Citation

  • Dominicy, Yves & Heikkilä, Matias & Ilmonen, Pauliina & Veredas, David, 2020. "Flexible multivariate Hill estimators," Journal of Econometrics, Elsevier, vol. 217(2), pages 398-410.
    • Handle: RePEc:eee:econom:v:217:y:2020:i:2:p:398-410
    DOI: 10.1016/j.jeconom.2019.12.010
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    References listed on IDEAS

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    1. Yves Dominicy & Pauliina Ilmonen & David Veredas, 2017. "Multivariate Hill Estimators," International Statistical Review, International Statistical Institute, vol. 85(1), pages 108-142, April.
    2. Qing Liu & Tiantian Mao & Taizhong Hu, 2017. "Closure properties of the second-order regular variation under convolutions," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(1), pages 104-119, January.
    3. Robert Serfling, 2010. "Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardisation," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(7), pages 915-936.
    4. Byczkowski, T. & Nolan, J. P. & Rajput, B., 1993. "Approximation of Multidimensional Stable Densities," Journal of Multivariate Analysis, Elsevier, vol. 46(1), pages 13-31, July.
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    Cited by:

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    More about this item

    Keywords

    Tail index; Hill estimator; Extreme value; Multivariate regular variation; Homogeneous function;
    All these keywords.

    JEL classification:

    • C01 - Mathematical and Quantitative Methods - - General - - - Econometrics
    • C39 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Other
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics
    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)

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