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Threshold selection and trimming in extremes.

Martin Bladt1, Hansjörg Albrecher2, Jan Beirlant3,4

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Summary
This summary is machine-generated.

This study introduces a novel method for extreme value statistics by trimming lower order statistics from the Hill estimator. This approach simplifies threshold selection for tail estimation, improving accuracy for insurance data analysis.

Keywords:
Hall classRegular variationThreshold selectionTrimming

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Area of Science:

  • Extreme Value Statistics
  • Statistical Inference
  • Risk Management

Background:

  • The classical Hill estimator is a cornerstone in extreme value statistics for tail estimation.
  • Accurate threshold selection is crucial but often challenging, hindering reliable tail characteristic estimation.
  • Existing methods can be sensitive to the choice of threshold, impacting the accuracy of extreme value analysis.

Purpose of the Study:

  • To develop a simplified and robust threshold selection procedure for the Hill estimator.
  • To introduce a novel trimming and rescaling technique for lower order statistics in extreme value analysis.
  • To propose an alternative tail index estimator with improved performance for lighter tails.

Main Methods:

  • Trimming lower order statistics from the Hill estimator and rescaling the remaining terms.
  • Analyzing the flatness of trimmed statistics trajectories near the optimal threshold.
  • Visual (trimmed Hill plots) and mathematical (minimizing expected empirical variance) threshold selection for the Hall class.

Main Results:

  • A simple threshold selection procedure for the Hill estimator is proposed, bypassing difficult tail characteristic estimations.
  • An alternative tail index estimator is derived, showing superior performance for lighter tails.
  • A ratio statistic routine is suggested for evaluating the quality of the selected threshold.

Conclusions:

  • The proposed trimming and rescaling method offers a straightforward and effective approach to threshold selection in extreme value statistics.
  • The new method demonstrates favorable performance in simulation studies and real-world insurance data analysis.
  • This technique enhances the reliability of tail estimation and risk assessment in practical applications.