Threshold selection and trimming in extremes
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Threshold selection and trimming in extremes ¨ Albrecher2 · Jan Beirlant3,4 Martin Bladt1 · Hansjorg Received: 9 December 2019 / Revised: 28 June 2020 / Accepted: 30 June 2020 / © The Author(s) 2020
Abstract We consider removing lower order statistics from the classical Hill estimator in extreme value statistics, and compensating for it by rescaling the remaining terms. Trajectories of these trimmed statistics as a function of the extent of trimming turn out to be quite flat near the optimal threshold value. For the regularly varying case, the classical threshold selection problem in tail estimation is then revisited, both visually via trimmed Hill plots and, for the Hall class, also mathematically via minimizing the expected empirical variance. This leads to a simple threshold selection procedure for the classical Hill estimator which circumvents the estimation of some of the tail characteristics, a problem which is usually the bottleneck in threshold selection. As a by-product, we derive an alternative estimator of the tail index, which assigns more weight to large observations, and works particularly well for relatively lighter tails. A simple ratio statistic routine is suggested to evaluate the goodness of the implied selection of the threshold. We illustrate the favourable performance and the potential of the proposed method with simulation studies and real insurance data. Keywords Trimming · Threshold selection · Regular variation · Hall class
Martin Bladt
[email protected] Hansj¨org Albrecher [email protected] Jan Beirlant [email protected] 1
Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, CH-1015, Lausanne, Switzerland
2
Department of Actuarial Science, Faculty of Business and Economics and Swiss Finance Institute, University of Lausanne, CH-1015, Lausanne, Switzerland
3
Department of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001, Leuven, Belgium
4
Department of Mathematical Statistics and Actuarial Science, University of the Free State, Bloemfontein, South Africa
M. Bladt et al.
AMS 2000 Subject Classifications 62F07 · 62P05
1 Introduction The use of Pareto-type tails has been shown to be important in different areas of risk management, such as for instance in computer science, insurance and finance. In social sciences and linguistics the model is referred to as Zipf’s law. This model corresponds to the max-domain of attraction of a generalized extreme value distribution with a positive extreme value index (EVI) ξ : 1 − F (x) = x −1/ξ (x),
ξ > 0,
(1)
where denotes a slowly varying function at infinity: (ux) = 1, for every u > 0. x→∞ (x) lim
(2)
Since the appearance of the paper of Hill (1975) in which the EVI estimator 1 log Xn−i+1,n − log Xn−k,n k k
Hk,n =
(3)
i=1
was proposed with Xn,n ≥ Xn−1,n ≥ · · · ≥ Xn−i+1,n ≥ · · · ≥ X1,n denoting the ordered statistics of a random sample from F , the literature on estimation of ξ > 0 and other tail quantities such as extreme quantiles and tail probabilities has inc
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