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.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Albrecher H, Beirlant J, Teugels J (2017) Reinsurance: actuarial and statistical aspects. Wiley, London
Acerbi C (2002) Spectral measures of risk: a coherent representation of subjective risk aversion. J Bank Finance 26:1505–1518
Acerbi C, Tasche D (2002) On the coherence of expected shortfall. J Bank Finance 26:1487–1503
Artzner P, Delbaen F, Eber J-M, Heath D (1999) Coherent measures of risk. Math Finance 9:203–228
Beirlant J, Goegebeur Y, Segers J, Teugels J (2004) Statistics of extremes: theory and applications. Wiley, London
Bellini F, Di Bernardino E (2015) Risk management with expectiles. Eur J Finance. https://doi.org/10.1080/1351847X.2015.1052150
Box GE, Cox DR (1964) An analysis of transformations. J R Stat Soc B 26:211–243
Breckling J, Chambers R (1988) M-quantiles. Biometrika 75:761–772
Chen Z (1996) Conditional \(L_p-\)quantiles and their application to testing of symmetry in non-parametric regression. Stat Probab Lett 29:107–115
Daouia A, Girard S, Stupfler G (2018) Estimation of tail risk based on extreme expectiles. J R Stat Soc B 80:263–292
Daouia A, Girard S, Stupfler G (2019) Extreme M-quantiles as risk measures: from \(L^1\) to \(L^p\) optimization. Bernoulli 25:264–309
Daouia A, Girard S, Stupfler G (2020) Tail expectile process and risk assessment. Bernoulli 26:531–556
Dekkers A, Einmhal J, de Haan L (1989) A moment estimator for the index of an extreme-value distribution. Ann Stat 17:1833–1855
de Haan L, Ferreira A (2006) Extreme value theory: an introduction. Springer series in operations research and financial engineering. Springer, London
El Methni J, Gardes L, Girard S (2018) Kernel estimation of extreme regression risk measures. Electron J Stat 12:359–398
El Methni J, Stupfler G (2017) Extreme versions of Wang risk measures and their estimation for heavy-tailed distributions. Stat Sin 27:907–930
El Methni J, Stupfler G (2018) Improved estimators of extreme Wang distortion risk measures for very heavy-tailed distributions. Econ Stat 6:129–148
Embrechts P, Kluppelberg C, Mikosch T (2013) Modelling extremal events for insurance and finance, vol 33. Springer, London
Furman E, Landsman Z (2006) Tail variance premium with applications for elliptical portfolio of risks. ASTIN Bull J IAA 36:433–462
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
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
Gómez-Déniz E, Calderín E (2014) Unconditional distributions obtained from conditional specifications models with applications in risk theory. Scand Act J 7:602–619
Gómez-Déniz E, Calderín E (2015) Modeling insurance data with the Pareto arctan distribution. ASTIN Bull 45(3):639–660
Hoeffding W (1948) A class of statistics with asymptotically normal distribution. Ann Math Stat 19(3):293–325
Holzmann H, Klar B (2016) Expectile asymptotics. Electron J Stat 10:2355–2371
Hua L, Joe H (2011) Second order regular variation and conditional tail expectation of multiple risks. Insur Math Econ 49:537–546
Jones MC (1994) Expectiles and M-quantiles are quantiles. Stat Probab Lett 20:149–153
Konstantinides D (2018) Risk theory. A heavy tail approach. World Scientific Publishing, Singapore
Leng C, Tong X (2014) Censored quantile regression via Box–Cox transformation under conditional independence. Stat Sin 24:221–249
McNeil AJ (1988) Estimating the tails of loss severity distribution using extreme value theory. ASTIN Bull 27(1):117–137
McNeil AJ, Frey R, Embrechts P (2005) Quantitative risk management. Princeton University Press, Princeton
Newey WK, Powell JL (1987) Asymmetric least squares estimation and testing. Econometrica 55:819–847
Resnick S (2001) Discussion of the Danish data on large fire insurance losses. ASTIN Bull 27(1):139–151
Resnick S (2007) Heavy-tail phenomena: probabilistic and statistical modeling. Springer, New York
Resnick S (2008) Extreme values, regular variation, and point Processes. Springer, London
Rockafellar RT, Uryasev S (2002) Conditional value-at-risk for general loss distributions. J Bank Finance 26:1443–1471
Rockafellar RT, Uryasev S, Zabarankin M (2006) Generalized deviations in risk analysis. Finance Stoch 10:51–74
Rolski T, Schmidli H, Schmidt V, Teugels JL (2009) Stochastic processes for insurance and finance, vol 505. Wiley, London
Tasche D (2002) Expected shortfall and beyond. J Bank Finance 26:1519–1533
Valdez EA (2005) Tail conditional variance for elliptically contoured distributions. Belg Act Bull 5:26–36
Weissman I (1978) Estimation of parameters and large quantiles based on the \(k\) largest observations. J Am Stat Assoc 73:812–815
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).
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
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
- Gini functional
- Risk measure
- Variability measure
- Distribution tail
- Extreme-value theory
Mathematics Subject Classification