• Bernoulli
  • Volume 25, Number 1 (2019), 264-309.

Extreme M-quantiles as risk measures: From $L^{1}$ to $L^{p}$ optimization

Abdelaati Daouia, Stéphane Girard, and Gilles Stupfler

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The class of quantiles lies at the heart of extreme-value theory and is one of the basic tools in risk management. The alternative family of expectiles is based on squared rather than absolute error loss minimization. It has recently been receiving a lot of attention in actuarial science, econometrics and statistical finance. Both quantiles and expectiles can be embedded in a more general class of M-quantiles by means of $L^{p}$ optimization. These generalized $L^{p}$-quantiles steer an advantageous middle course between ordinary quantiles and expectiles without sacrificing their virtues too much for $1<p<2$. In this paper, we investigate their estimation from the perspective of extreme values in the class of heavy-tailed distributions. We construct estimators of the intermediate $L^{p}$-quantiles and establish their asymptotic normality in a dependence framework motivated by financial and actuarial applications, before extrapolating these estimates to the very far tails. We also investigate the potential of extreme $L^{p}$-quantiles as a tool for estimating the usual quantiles and expectiles themselves. We show the usefulness of extreme $L^{p}$-quantiles and elaborate the choice of $p$ through applications to some simulated and financial real data.

Article information

Bernoulli, Volume 25, Number 1 (2019), 264-309.

Received: November 2016
Revised: May 2017
First available in Project Euclid: 12 December 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

asymptotic normality dependent observations expectiles extrapolation extreme values heavy tails $L^{p}$ optimization mixing quantiles tail risk


Daouia, Abdelaati; Girard, Stéphane; Stupfler, Gilles. Extreme M-quantiles as risk measures: From $L^{1}$ to $L^{p}$ optimization. Bernoulli 25 (2019), no. 1, 264--309. doi:10.3150/17-BEJ987.

Export citation


  • [1] Abdous, B. and Rémillard, B. (1995). Relating quantiles and expectiles under weighted-symmetry. Ann. Inst. Statist. Math. 47 371–384.
  • [2] Acharya, V.V., Pedersen, L.H., Philippon, T. and Richardson, M. (2012). Measuring systemic risk. Discussion Paper DP8824, Centre for Economic Policy Research, London.
  • [3] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk. Math. Finance 9 203–228.
  • [4] Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. (2004). Statistics of Extremes. Theory and Applications. Wiley Series in Probability and Statistics. Chichester: Wiley. With contributions from Daniel De Waal and Chris Ferro.
  • [5] Bellini, F. and Di Bernardino, E. (2017). Risk management with expectiles. Eur. J. Finance 23 487–506. DOI:10.1080/1351847X.2015.1052150.
  • [6] Bellini, F., Klar, B., Müller, A. and Rosazza Gianin, E. (2014). Generalized quantiles as risk measures. Insurance Math. Econom. 54 41–48.
  • [7] Boente, G. and Fraiman, R. (1995). Asymptotic distribution of smoothers based on local means and local medians under dependence. J. Multivariate Anal. 57 77–90.
  • [8] Bradley, R.C. (1988). A central limit theorem for stationary $\rho$-mixing sequences with infinite variance. Ann. Probab. 16 313–332.
  • [9] Bradley, R.C. (2005). Basic properties of strong mixing conditions. A survey and some open questions. Probab. Surv. 2 107–144. Update of, and a supplement to, the 1986 original.
  • [10] Breckling, J. and Chambers, R. (1988). $M$-quantiles. Biometrika 75 761–771.
  • [11] Brownlees, C.T. and Engle, R. (2012). Volatility, correlation and tails for systemic risk measurement. Preprint.
  • [12] Cai, J.-J., Einmahl, J.H.J., de Haan, L. and Zhou, C. (2015). Estimation of the marginal expected shortfall: The mean when a related variable is extreme. J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 417–442.
  • [13] Cai, Z. (2003). Nonparametric estimation equations for time series data. Statist. Probab. Lett. 62 379–390.
  • [14] Chavez-Demoulin, V., Embrechts, P. and Sardy, S. (2014). Extreme-quantile tracking for financial time series. J. Econometrics 181 44–52.
  • [15] Chen, Z. (1996). Conditional $L_{p}$-quantiles and their application to the testing of symmetry in non-parametric regression. Statist. Probab. Lett. 29 107–115.
  • [16] Daouia, A., Florens, J.-P. and Simar, L. (2010). Frontier estimation and extreme value theory. Bernoulli 16 1039–1063.
  • [17] Daouia, A., Gardes, L. and Girard, S. (2013). On kernel smoothing for extremal quantile regression. Bernoulli 19 2557–2589.
  • [18] Daouia, A., Girard, S. and Stupfler, G. (2017). Estimation of tail risk based on extreme expectiles. J. R. Stat. Soc. Ser. B. Stat. Methodol. To appear, DOI:10.1111/rssb.12254.
  • [19] Daouia, A., Girard, S. and Stupfler, G. (2017). Supplement to “Extreme M-quantiles as risk measures: From $L^{1}$ to $L^{p}$ optimization.” DOI:10.3150/17-BEJ987SUPP.
  • [20] Davis, R. and Mikosch, T. (2009). The extremogram: A correlogram for extreme events. Bernoulli 15 977–1009.
  • [21] Davis, R., Mikosch, T. and Cribben, I. (2012). Towards estimating extremal serial dependence via the bootstrapped extremogram. J. Econometrics 170 142–152.
  • [22] Davis, R.A., Mikosch, T. and Zhao, Y. (2013). Measures of serial extremal dependence and their estimation. Stochastic Process. Appl. 123 2575–2602.
  • [23] de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. New York: Springer.
  • [24] de Haan, L., Mercadier, C. and Zhou, C. (2016). Adapting extreme value statistics to financial time series: Dealing with bias and serial dependence. Finance Stoch. 20 321–354.
  • [25] Dekkers, A.L.M., Einmahl, J.H.J. and de Haan, L. (1989). A moment estimator for the index of an extreme-value distribution. Ann. Statist. 17 1833–1855.
  • [26] Drees, H. (2000). Weighted approximations of tail processes for $\beta$-mixing random variables. Ann. Appl. Probab. 10 1274–1301.
  • [27] Drees, H. (2002). Tail empirical processes under mixing conditions. In Empirical Processes Techniques for Dependent Data (H.G. Dehling, T. Mikosch and M. Sørensen, eds.). Boston: Birkhäuser.
  • [28] Drees, H. (2003). Extreme quantile estimation for dependent data, with applications to finance. Bernoulli 9 617–657.
  • [29] Drees, H. and Rootzén, H. (2010). Limit theorems for empirical processes of cluster functionals. Ann. Statist. 38 2145–2186.
  • [30] Efron, B. (1991). Regression percentiles using asymmetric squared error loss. Statist. Sinica 1 93–125.
  • [31] El Methni, J., Gardes, L. and Girard, S. (2014). Nonparametric estimation of extreme risks from conditional heavy-tailed distributions. Scand. J. Stat. 41 988–1012.
  • [32] El Methni, J. and Stupfler, G. (2017). Extreme versions of Wang risk measures and their estimation for heavy-tailed distributions. Statist. Sinica 27 907–930.
  • [33] El Methni, J. and Stupfler, G. (2017). Improved estimators of extreme Wang distortion risk measures for very heavy-tailed distributions. Econom. Stat. To appear. DOI:10.1016/j.ecosta.2017.03.002.
  • [34] Embrechts, P. and Hofert, M. (2014). Statistics and quantitative risk management for banking and insurance. Ann. Rev. Stat. Appl. 1 493–514.
  • [35] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Berlin: Springer.
  • [36] Gardes, L. and Stupfler, G. (2014). Estimation of the conditional tail index using a smoothed local Hill estimator. Extremes 17 45–75.
  • [37] Geyer, C.J. (1996). On the asymptotics of convex stochastic optimization. Unpublished manuscript.
  • [38] Gneiting, T. (2011). Making and evaluating point forecasts. J. Amer. Statist. Assoc. 106 746–762.
  • [39] Goegebeur, Y., Guillou, A. and Osmann, M. (2014). A local moment type estimator for the extreme value index in regression with random covariates. Canad. J. Statist. 42 487–507.
  • [40] Gong, J., Li, Y., Peng, L. and Yao, Q. (2015). Estimation of extreme quantiles for functions of dependent random variables. J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 1001–1024.
  • [41] Gudendorf, G. and Segers, J. (2010). Extreme-value copulas. In Copula Theory and Its Applications. Lect. Notes Stat. Proc. 198 127–145. Springer, Heidelberg.
  • [42] Hill, B.M. (1975). A simple general approach to inference about the tail of a distribution. Ann. Statist. 3 1163–1174.
  • [43] Honda, T. (2000). Nonparametric estimation of a conditional quantile for $\alpha$-mixing processes. Ann. Inst. Statist. Math. 52 459–470.
  • [44] Hsing, T. (1991). On tail index estimation using dependent data. Ann. Statist. 19 1547–1569.
  • [45] Hua, L. and Joe, H. (2011). Second order regular variation and conditional tail expectation of multiple risks. Insurance Math. Econom. 49 537–546.
  • [46] Ibragimov, I.A. (1975). A remark on the central limit theorem for dependent random variables. Theor. Probab. Appl. 20 135–141.
  • [47] Jones, M.C. (1994). Expectiles and $M$-quantiles are quantiles. Statist. Probab. Lett. 20 149–153.
  • [48] Knight, K. (1999). Epi-convergence in distribution and stochastic equi-semicontinuity. Technical report, Univ. Toronto.
  • [49] Koenker, R. and Bassett, G.S. (1978). Regression quantiles. Econometrica 46 33–50.
  • [50] Kuan, C-M., Yeh, J-H. and Hsu, Y-C. (2009). Assessing value at risk with CARE, the conditional autoregressive expectile models. J. Econometrics 2 261–270.
  • [51] Mao, T., Ng, K.W. and Hu, T. (2015). Asymptotic expansions of generalized quantiles and expectiles for extreme risks. Probab. Engrg. Inform. Sci. 29 309–327.
  • [52] Mao, T. and Yang, F. (2015). Risk concentration based on expectiles for extreme risks under FGM copula. Insurance Math. Econom. 64 429–439.
  • [53] Nelsen, R.B. (2006). An Introduction to Copulas, 2nd ed. New York: Springer.
  • [54] Newey, W.K. and Powell, J.L. (1987). Asymmetric least squares estimation and testing. Econometrica 55 819–847.
  • [55] Peligrad, M. (1987). On the central limit theorem for $\rho$-mixing sequences of random variables. Ann. Probab. 15 1387–1394.
  • [56] Pickands, J. (1975). Statistical inference using extreme order statistics. Ann. Statist. 3 119–131.
  • [57] Resnick, S. and Stărică, C. (1997). Asymptotic behavior of Hill’s estimator for autoregressive data. Commun. Stat., Stochastic Models 13 703–721.
  • [58] Resnick, S. and Stărică, C. (1998). Tail index estimation for dependent data. Ann. Appl. Probab. 8 1156–1183.
  • [59] Resnick, S. and Stǎricǎ, C. (1995). Consistency of Hill’s estimator for dependent data. J. Appl. Probab. 32 139–167.
  • [60] Resnick, S.I. (2007). Heavy-Tail Phenomena. Probabilistic and Statistical Modeling. Springer Series in Operations Research and Financial Engineering. New York: Springer.
  • [61] Robert, C.Y. (2008). Estimating the multivariate extremal index function. Bernoulli 14 1027–1064.
  • [62] Robert, C.Y. (2009). Inference for the limiting cluster size distribution of extreme values. Ann. Statist. 37 271–310.
  • [63] Rootzén, H. (2009). Weak convergence of the tail empirical process for dependent sequences. Stochastic Process. Appl. 119 468–490.
  • [64] Schnabel, S. and Eilers, P. (2013). A location-scale model for non-crossing expectile curves. Stat 2 171–183. DOI:10.1002/sta4.27.
  • [65] Schulze Waltrup, L., Sobotka, F., Kneib, T. and Kauermann, G. (2015). Expectile and quantile regression – David and Goliath? Stat. Model. 15 433–456.
  • [66] Sklar, M. (1959). Fonctions de répartition à $n$ dimensions et leurs marges. Publ. Inst. Stat. Univ. Paris 8 229–231.
  • [67] Stupfler, G. (2013). A moment estimator for the conditional extreme-value index. Electron. J. Stat. 7 2298–2343.
  • [68] Utev, S.A. (1990). On the central limit theorem for $\varphi$-mixing arrays of random variables. Theory Probab. Appl. 35 131–139.
  • [69] Weissman, I. (1978). Estimation of parameters and large quantiles based on the $k$ largest observations. J. Amer. Statist. Assoc. 73 812–815.
  • [70] Yao, Q. and Tong, H. (1996). Asymmetric least squares regression estimation: A nonparametric approach. J. Nonparametr. Stat. 6 273–292.
  • [71] Zhao, G.H., Teo, K.L. and Chan, K.S. (2005). Estimation of conditional quantiles by a new smoothing approximation of asymmetric loss functions. Stat. Comput. 15 5–11.
  • [72] Ziegel, J.F. (2016). Coherence and elicitability. Math. Finance 26 901–918.
  • [73] Zou, H. (2014). Generalizing Koenker’s distribution. J. Statist. Plann. Inference 148 123–127.

Supplemental materials

  • Supplement to “Extreme M-quantiles as risk measures: From $L^{1}$ to $L^{p}$ optimization.”. The supplement to this article contains additional simulations, a second application to medical insurance data, technical lemmas and the proofs of all theoretical results of the main article.