On the estimation of the variability in the distribution tail

Abstract

We propose a new measure of variability in the tail of a distribution by applying a Box–Cox transformation of parameter \(p \ge 0\) to the tail-Gini functional. It is shown that the so-called Box–Cox Tail Gini Variability measure is a valid variability measure whose condition of existence may be as weak as necessary thanks to the tuning parameter p. The tail behaviour of the measure is investigated under a general extreme-value condition on the distribution tail. We then show how to estimate the Box–Cox Tail Gini Variability measure within the range of the data. These methods provide us with basic estimators that are then extrapolated using the extreme-value assumption to estimate the variability in the very far tails. The finite sample behaviour of the estimators is illustrated both on simulated and real data.

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Acknowledgements

The authors acknowledge two anonymous reviewers for their helpful comments that led to an improved version of this paper. This research was supported by the French National Research Agency under the Grant ANR-19-CE40-0013-01/ExtremReg project. S. Girard gratefully acknowledges the support of the Chair Stress Test, Risk Management and Financial Steering, led by the French Ecole Polytechnique and its Foundation and sponsored by BNP Paribas, as well as the support of the French National Research Agency in the framework of the Investissements d’Avenir program (ANR-15-IDEX-02).

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Correspondence to Laurent Gardes.

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Gardes, L., Girard, S. On the estimation of the variability in the distribution tail. TEST (2021). https://doi.org/10.1007/s11749-021-00754-2

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Keywords

  • Gini functional
  • Risk measure
  • Variability measure
  • Distribution tail
  • Extreme-value theory

Mathematics Subject Classification

  • 62G32
  • 62G20