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Weak properties and robustness of t-Hill estimators

Abstract

We describe a novel method of heavy tails estimation based on transformed score (t-score). Based on a new score moment method we derive the t-Hill estimator, which estimates the extreme value index of a distribution function with regularly varying tail. t-Hill estimator is distribution sensitive, thus it differs in e.g. Pareto and log-gamma case. Here, we study both forms of the estimator, i.e. t-Hill and t-lgHill. For both estimators we prove weak consistency in moving average settings as well as the asymptotic normality of t-lgHill estimator in iid setting. In cases of contamination with heavier tails than the tail of original sample, t-Hill outperforms several robust tail estimators, especially in small samples. A simulation study emphasizes the fact that the level of contamination is playing a crucial role. The larger the contamination, the better are the t-score moment estimates. The reason for this is the bounded t-score of heavy-tailed distributions (and, consequently, bounded influence functions of the estimators). We illustrate the developed methodology on a small sample data set of stake measurements from Guanaco glacier in Chile.

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Correspondence to Milan Stehlík.

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Jordanova, P., Fabián, Z., Hermann, P. et al. Weak properties and robustness of t-Hill estimators. Extremes 19, 591–626 (2016). https://doi.org/10.1007/s10687-016-0256-2

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Keywords

  • Point estimation
  • Asymptotic properties of estimators
  • t-Hill estimator
  • t-lgHill estimator

AMS 2000 Subject Classifications

  • 62F10
  • 62F12