, Volume 19, Issue 4, pp 591–626 | Cite as

Weak properties and robustness of t-Hill estimators

  • Pavlina Jordanova
  • Zdeněk Fabián
  • Philipp Hermann
  • Luboš Střelec
  • Andrés Rivera
  • Stéphane Girard
  • Sebastián Torres
  • Milan StehlíkEmail author
Open Access


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.


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

AMS 2000 Subject Classifications

62F10 62F12 


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© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Pavlina Jordanova
    • 1
  • Zdeněk Fabián
    • 2
  • Philipp Hermann
    • 3
  • Luboš Střelec
    • 4
  • Andrés Rivera
    • 5
    • 6
  • Stéphane Girard
    • 7
  • Sebastián Torres
    • 7
  • Milan Stehlík
    • 8
    • 9
    Email author
  1. 1.Faculty of Mathematics and InformaticsShumen UniversityShumenBulgaria
  2. 2.Institute of Computer ScienceAcademy of Sciences of the Czech RepublicPragueCzech Republic
  3. 3.Department of Applied StatisticsJohannes Kepler University LinzLinzAustria
  4. 4.Department of Statistics and Operation Analysis (FBE)Mendel University in BrnoBrnoCzech Republic
  5. 5.Laboratorio de GlaciologíaCentro de Estudios Científicos (CECs)ValdiviaChile
  6. 6.Departamento de GeografíaUniversidad de ChileSantiagoChile
  7. 7.INRIA Rhône-Alpes, team Mistis, InovalléeMontbonnotFrance
  8. 8.Institute of StatisticsUniversidad de ValparaísoValparaísoChile
  9. 9.Department of Applied StatisticsJohannes Kepler University LinzLinzAustria

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