Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-29T04:16:51.628Z Has data issue: false hasContentIssue false
  • English
  • Français

Scaling of High-Quantile Estimators

Part of: Stochastic processes Nonparametric inference

Published online by Cambridge University Press:  14 July 2016

Matthias Degen  and
Paul Embrechts
Matthias Degen*
Affiliation:
ETH Zürich
Paul Embrechts*
Affiliation:
ETH Zürich and Swiss Finance Institute
*
∗ Postal address: Department of Mathematics, ETH Zürich, Raemistrasse 101, CH-8092 Zürich, Switzerland.
∗ Postal address: Department of Mathematics, ETH Zürich, Raemistrasse 101, CH-8092 Zürich, Switzerland.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Extreme value theorypeaks over thresholdpenultimate approximationpower normalizationsecond-order extended regular variation

MSC classification

Secondary: 60G70: Extreme value theory; extremal processes 62G32: Statistics of extreme values; tail inference
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 4 , December 2011 , pp. 968 - 983
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Balkema, G. and Embrechts, P. (2007). High Risk Scenarios and Extremes. A geometric approach. European Mathematical Society, Zürich.Google Scholar
[2] 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.Google Scholar
[3] Basel Committee on Banking Supervision (2008). Guidelines for computing capital for incremental risk in the trading book. Bank for International Settlements, Basel.Google Scholar
[4] Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. (2004). Statistics of Extremes. John Wiley, Chichester.Google Scholar
[5] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
[6] Chavez-Demoulin, V. and Embrechts, P. (2011). An EVT primer for credit risk. In The Oxford Handbook of Credit Derivatives, eds Lipton, A. and Rennie, A., Oxford University Press, pp. 500–532.Google Scholar
[7] Cohen, J. P. (1982). Convergence rates for the ultimate and penultimate approximations in extreme-value theory. Adv. Appl. Prob. 14, 833–854.Google Scholar
[8] Crouhy, M., Galai, D. and Mark, R. (2006). The Essentials of Risk Management. McGraw-Hill, New York.Google Scholar
[9] Daníelsson, J. et al. (2001). An academic response to Basel II. Financial Markets Group, London School of Economics.Google Scholar
[10] de Haan, L. (1970). On Regular Variation and Its Applications to the Weak Convergence of Sample Extremes (Math. Centre Tracts 32), Mathematisch Centrum, Amsterdam.Google Scholar
[11] de Haan, L. and Ferreira, A. (2006). Extreme Value Theory. Springer, New York.Google Scholar
[12] Embrechts, P. and Hofert, M. (2010). A note on generalized inverses. Preprint, ETH Zürich.Google Scholar
[13] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.Google Scholar
[14] 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.Google Scholar
[15] Gnedenko, B. (1943). Sur la distribution limite du terme maximum d'une série aléatoire. Ann. Math. 44, 423–453.Google Scholar
[16] Gomes, M. I. and de Haan, L. (1999). Approximation by penultimate extreme value distributions. Extremes 2, 71–85.Google Scholar
[17] Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences. Wiley, Hoboken, NJ.Google Scholar
[18] Loader, C. (1999). Local Regression and Likelihood. Springer, New York.Google Scholar
[19] 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.Google Scholar
[20] 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.Google Scholar
[21] Nešlehová, J., Embrechts, P. and Chavez-Demoulin, V. (2006). Infinite mean models and the LDA for operational risk. J. Operat. Risk 1, 3–25.Google Scholar
[22] 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.Google Scholar
[23] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.Google Scholar
[24] Tukey, J. W. (1977). Exploratory Data Analysis. Addison-Wesley, Reading.Google Scholar