A robust estimator for the tail index of Pareto-type distributions |
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Authors: | B. Vandewalle J. Beirlant A. Christmann |
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Affiliation: | a Department of Mathematics, Katholieke Universiteit Leuven, Belgium b Center for Statistics and Applications, Universidade de Lisboa, Portugal c Department of Mathematics and University Center of Statistics, W. de Croylaan 54, 3000-Leuven, Katholieke Universiteit Leuven, Belgium d Department of Mathematics, Vrije Universiteit Brussel, Belgium |
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Abstract: | In extreme value statistics, the extreme value index is a well-known parameter to measure the tail heaviness of a distribution. Pareto-type distributions, with strictly positive extreme value index (or tail index) are considered. The most prominent extreme value methods are constructed on efficient maximum likelihood estimators based on specific parametric models which are fitted to excesses over large thresholds. Maximum likelihood estimators however are often not very robust, which makes them sensitive to few particular observations. Even in extreme value statistics, where the most extreme data usually receive most attention, this can constitute a serious problem. The problem is illustrated on a real data set from geopedology, in which a few abnormal soil measurements highly influence the estimates of the tail index. In order to overcome this problem, a robust estimator of the tail index is proposed, by combining a refinement of the Pareto approximation for the conditional distribution of relative excesses over a large threshold with an integrated squared error approach on partial density component estimation. It is shown that the influence function of this newly proposed estimator is bounded and through several simulations it is illustrated that it performs reasonably well at contaminated as well as uncontaminated data. |
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Keywords: | Extreme value statistics Relative excesses over a large threshold Robust tail index estimation Geopedology |
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