Mean-of-order p reduced-bias extreme value index estimation under a third-order framework

Abstract

Reduced-bias versions of a very simple generalization of the ‘classical’ Hill estimator of a positive extreme value index (EVI) are put forward. The Hill estimator can be regarded as the logarithm of the mean-of-order-0 of a certain set of statistics. Instead of such a geometric mean, it is sensible to consider the mean-of-order-p (MOP) of those statistics, with p real. Under a third-order framework, the asymptotic behaviour of the MOP, optimal MOP and associated reduced-bias classes of EVI-estimators is derived. Information on the dominant non-null asymptotic bias is also provided so that we can deal with an asymptotic comparison at optimal levels of some of those classes. Large-scale Monte-Carlo simulation experiments are undertaken to provide finite sample comparisons.

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Correspondence to Frederico Caeiro.

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Caeiro, F., Gomes, M.I., Beirlant, J. et al. Mean-of-order p reduced-bias extreme value index estimation under a third-order framework. Extremes 19, 561–589 (2016). https://doi.org/10.1007/s10687-016-0261-5

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Keywords

  • Bias estimation
  • Heavy tails
  • Optimal levels
  • Semi-parametric reduced-bias estimation
  • Statistics of extremes

AMS 2000 Subject Classifications

  • Primary 62G32
  • Secondary 65C05