Weak properties and robustness of t-Hill estimators |
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Authors: | Pavlina Jordanova Zdeněk Fabián Philipp Hermann Luboš Střelec Andrés Rivera Stéphane Girard Sebastián Torres Milan Stehlík |
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Institution: | 1.Faculty of Mathematics and Informatics,Shumen University,Shumen,Bulgaria;2.Institute of Computer Science,Academy of Sciences of the Czech Republic,Prague,Czech Republic;3.Department of Applied Statistics,Johannes Kepler University Linz,Linz,Austria;4.Department of Statistics and Operation Analysis (FBE),Mendel University in Brno,Brno,Czech Republic;5.Laboratorio de Glaciología,Centro de Estudios Científicos (CECs),Valdivia,Chile;6.Departamento de Geografía,Universidad de Chile,Santiago,Chile;7.INRIA Rh?ne-Alpes, team Mistis, Inovallée,Montbonnot,France;8.Institute of Statistics,Universidad de Valparaíso,Valparaíso,Chile;9.Department of Applied Statistics,Johannes Kepler University Linz,Linz,Austria |
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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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