Journal of Applied Probability

Scaling of high-quantile estimators

Matthias Degen and Paul Embrechts

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


Enhanced by the global financial crisis, the discussion about an accurate estimation of regulatory (risk) capital a financial institution needs to hold in order to safeguard against unexpected losses has become highly relevant again. The presence of heavy tails in combination with small sample sizes turns estimation at such extreme quantile levels into an inherently difficult statistical issue. We discuss some of the problems and pitfalls that may arise. In particular, based on the framework of second-order extended regular variation, we compare different high-quantile estimators and propose methods for the improvement of standard methods by focusing on the concept of penultimate approximations.

Article information

J. Appl. Probab., Volume 48, Number 4 (2011), 968-983.

First available in Project Euclid: 16 December 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 62G32: Statistics of extreme values; tail inference

Extreme value theory peaks over threshold penultimate approximation power normalization second-order extended regular variation


Degen, Matthias; Embrechts, Paul. Scaling of high-quantile estimators. J. Appl. Probab. 48 (2011), no. 4, 968--983. doi:10.1239/jap/1324046013.

Export citation


  • Balkema, G. and Embrechts, P. (2007). High Risk Scenarios and Extremes. A geometric approach. European Mathematical Society, Zürich.
  • Barakat, H. M., Nigm, E. M. and El-Adll, M. E. (2010). Comparison between the rates of convergence of extremes under linear and under power normalization. Statist. Papers 51, 149–164.
  • Basel Committee on Banking Supervision (2008). Guidelines for computing capital for incremental risk in the trading book. Bank for International Settlements, Basel.
  • Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. (2004). Statistics of Extremes. John Wiley, Chichester.
  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.
  • Chavez-Demoulin, V. and Embrechts, P. (2011). An EVT primer for credit risk. In The Oxford Handbook of Credit Derivatives, eds A. Lipton and A. Rennie, Oxford University Press, pp. 500–532.
  • Cohen, J. P. (1982). Convergence rates for the ultimate and penultimate approximations in extreme-value theory. Adv. Appl. Prob. 14, 833–854.
  • Crouhy, M., Galai, D. and Mark, R. (2006). The Essentials of Risk Management. McGraw-Hill, New York.
  • Daníelsson, J. et al. (2001). An academic response to Basel II. Financial Markets Group, London School of Economics.
  • de Haan, L. (1970). On Regular Variation and Its Applications to the Weak Convergence of Sample Extremes (Math. Centre Tracts 32), Mathematisch Centrum, Amsterdam.
  • de Haan, L. and Ferreira, A. (2006). Extreme Value Theory. Springer, New York.
  • Embrechts, P. and Hofert, M. (2010). A note on generalized inverses. Preprint, ETH Zürich.
  • Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.
  • Fisher, R. A. and Tippett, L. H. C. (1928). Limiting forms of the frequency distribution of the largest or smallest member of a sample. Proc. Camb. Phil. Soc. 24, 180–190.
  • Gnedenko, B. (1943). Sur la distribution limite du terme maximum d'une série aléatoire. Ann. Math. 44, 423–453.
  • Gomes, M. I. and de Haan, L. (1999). Approximation by penultimate extreme value distributions. Extremes 2, 71–85.
  • Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences. Wiley, Hoboken, NJ.
  • Loader, C. (1999). Local Regression and Likelihood. Springer, New York.
  • Mohan, N. R. and Ravi, S. (1992). Max domains of attraction of univariate and multivariate $p$-max stable laws. Theoret. Prob. Appl. 37, 632–643.
  • Moscadelli, M. (2004). The modelling of operational risk: experiences with the analysis of the data collected by the Basel Committee. Working Paper No 517, Bank of Italy.
  • Nešlehová, J., Embrechts, P. and Chavez-Demoulin, V. (2006). Infinite mean models and the \textnormalLDA for operational risk. J. Operat. Risk 1, 3–25.
  • Pantcheva, E. (1985). Limit theorems for extreme order statistics under nonlinear normalization. In Stability Problems for Stochastic Models (Uzhgorod, 1984; Lecture Notes Math. 1155), Springer, Berlin, pp. 284–309.
  • Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
  • Tukey, J. W. (1977). Exploratory Data Analysis. Addison-Wesley, Reading.