On the estimation of the variability in the distribution tail

TEST (2021)Cite this article

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.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

US$ 39.95

Tax calculation will be finalised during checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

References

  1. Albrecher H, Beirlant J, Teugels J (2017) Reinsurance: actuarial and statistical aspects. Wiley, London

    Google Scholar 

  2. Acerbi C (2002) Spectral measures of risk: a coherent representation of subjective risk aversion. J Bank Finance 26:1505–1518

    Article  Google Scholar 

  3. Acerbi C, Tasche D (2002) On the coherence of expected shortfall. J Bank Finance 26:1487–1503

    Article  Google Scholar 

  4. Artzner P, Delbaen F, Eber J-M, Heath D (1999) Coherent measures of risk. Math Finance 9:203–228

    MathSciNet  Article  Google Scholar 

  5. Beirlant J, Goegebeur Y, Segers J, Teugels J (2004) Statistics of extremes: theory and applications. Wiley, London

    Google Scholar 

  6. Bellini F, Di Bernardino E (2015) Risk management with expectiles. Eur J Finance. https://doi.org/10.1080/1351847X.2015.1052150

    Article  Google Scholar 

  7. Box GE, Cox DR (1964) An analysis of transformations. J R Stat Soc B 26:211–243

    MATH  Google Scholar 

  8. Breckling J, Chambers R (1988) M-quantiles. Biometrika 75:761–772

    MathSciNet  Article  Google Scholar 

  9. Chen Z (1996) Conditional \(L_p-\)quantiles and their application to testing of symmetry in non-parametric regression. Stat Probab Lett 29:107–115

    Article  Google Scholar 

  10. Daouia A, Girard S, Stupfler G (2018) Estimation of tail risk based on extreme expectiles. J R Stat Soc B 80:263–292

    MathSciNet  Article  Google Scholar 

  11. Daouia A, Girard S, Stupfler G (2019) Extreme M-quantiles as risk measures: from \(L^1\) to \(L^p\) optimization. Bernoulli 25:264–309

    MathSciNet  Article  Google Scholar 

  12. Daouia A, Girard S, Stupfler G (2020) Tail expectile process and risk assessment. Bernoulli 26:531–556

    MathSciNet  Article  Google Scholar 

  13. Dekkers A, Einmhal J, de Haan L (1989) A moment estimator for the index of an extreme-value distribution. Ann Stat 17:1833–1855

    MathSciNet  Article  Google Scholar 

  14. de Haan L, Ferreira A (2006) Extreme value theory: an introduction. Springer series in operations research and financial engineering. Springer, London

    Google Scholar 

  15. El Methni J, Gardes L, Girard S (2018) Kernel estimation of extreme regression risk measures. Electron J Stat 12:359–398

    MathSciNet  Article  Google Scholar 

  16. El Methni J, Stupfler G (2017) Extreme versions of Wang risk measures and their estimation for heavy-tailed distributions. Stat Sin 27:907–930

    MathSciNet  MATH  Google Scholar 

  17. El Methni J, Stupfler G (2018) Improved estimators of extreme Wang distortion risk measures for very heavy-tailed distributions. Econ Stat 6:129–148

    MathSciNet  Google Scholar 

  18. Embrechts P, Kluppelberg C, Mikosch T (2013) Modelling extremal events for insurance and finance, vol 33. Springer, London

    Google Scholar 

  19. Furman E, Landsman Z (2006) Tail variance premium with applications for elliptical portfolio of risks. ASTIN Bull J IAA 36:433–462

    MathSciNet  Article  Google Scholar 

  20. Furman E, Wang R, Zitikis R (2017) Gini-type measures of risk and variability: Gini shortfall, capital allocations, and heavy-tailed risks. J Bank Finance 83:70–84

    Article  Google Scholar 

  21. Gardes L, Girard S, Stupfler G (2020) Beyond tail median and conditional tail expectation: extreme risk estimation using tail \(L^p-\)optimisation. Scand J Stat 47:922–949

    MathSciNet  Article  Google Scholar 

  22. Gómez-Déniz E, Calderín E (2014) Unconditional distributions obtained from conditional specifications models with applications in risk theory. Scand Act J 7:602–619

    MathSciNet  Article  Google Scholar 

  23. Gómez-Déniz E, Calderín E (2015) Modeling insurance data with the Pareto arctan distribution. ASTIN Bull 45(3):639–660

    MathSciNet  Article  Google Scholar 

  24. Hoeffding W (1948) A class of statistics with asymptotically normal distribution. Ann Math Stat 19(3):293–325

    MathSciNet  Article  Google Scholar 

  25. Holzmann H, Klar B (2016) Expectile asymptotics. Electron J Stat 10:2355–2371

    MathSciNet  Article  Google Scholar 

  26. Hua L, Joe H (2011) Second order regular variation and conditional tail expectation of multiple risks. Insur Math Econ 49:537–546

    MathSciNet  Article  Google Scholar 

  27. Jones MC (1994) Expectiles and M-quantiles are quantiles. Stat Probab Lett 20:149–153

    MathSciNet  Article  Google Scholar 

  28. Konstantinides D (2018) Risk theory. A heavy tail approach. World Scientific Publishing, Singapore

    Google Scholar 

  29. Leng C, Tong X (2014) Censored quantile regression via Box–Cox transformation under conditional independence. Stat Sin 24:221–249

    MathSciNet  MATH  Google Scholar 

  30. McNeil AJ (1988) Estimating the tails of loss severity distribution using extreme value theory. ASTIN Bull 27(1):117–137

    Article  Google Scholar 

  31. McNeil AJ, Frey R, Embrechts P (2005) Quantitative risk management. Princeton University Press, Princeton

    Google Scholar 

  32. Newey WK, Powell JL (1987) Asymmetric least squares estimation and testing. Econometrica 55:819–847

    MathSciNet  Article  Google Scholar 

  33. Resnick S (2001) Discussion of the Danish data on large fire insurance losses. ASTIN Bull 27(1):139–151

    Article  Google Scholar 

  34. Resnick S (2007) Heavy-tail phenomena: probabilistic and statistical modeling. Springer, New York

    Google Scholar 

  35. Resnick S (2008) Extreme values, regular variation, and point Processes. Springer, London

    Google Scholar 

  36. Rockafellar RT, Uryasev S (2002) Conditional value-at-risk for general loss distributions. J Bank Finance 26:1443–1471

    Article  Google Scholar 

  37. Rockafellar RT, Uryasev S, Zabarankin M (2006) Generalized deviations in risk analysis. Finance Stoch 10:51–74

    MathSciNet  Article  Google Scholar 

  38. Rolski T, Schmidli H, Schmidt V, Teugels JL (2009) Stochastic processes for insurance and finance, vol 505. Wiley, London

    Google Scholar 

  39. Tasche D (2002) Expected shortfall and beyond. J Bank Finance 26:1519–1533

    Article  Google Scholar 

  40. Valdez EA (2005) Tail conditional variance for elliptically contoured distributions. Belg Act Bull 5:26–36

    MATH  Google Scholar 

  41. Weissman I (1978) Estimation of parameters and large quantiles based on the \(k\) largest observations. J Am Stat Assoc 73:812–815

    MathSciNet  MATH  Google Scholar 

Download references

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).

Author information

Affiliations

  1. Institut de Recherche Mathématique Avancée, UMR 7501 Université de Strasbourg et CNRS, 7 rue René Descartes, 67000, Strasbourg, France

    Laurent Gardes

  2. Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, 38000, Grenoble, France

    Stéphane Girard

Authors
  1. Laurent Gardes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stéphane Girard
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Laurent Gardes.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Keywords

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

Mathematics Subject Classification

  • 62G32
  • 62G20