Reduced-Bias Estimator of the Conditional Tail Expectation of Heavy-Tailed Distributions

  • El Hadji Deme
  • Stéphane Girard
  • Armelle GuillouEmail author


Several risk measures have been proposed in the literature. In this paper, we focus on the estimation of the Conditional Tail Expectation (CTE). Its asymptotic normality has been first established in the literature under the classical assumption that the second moment of the loss variable is finite, this condition being very restrictive in practical applications. Such a result has been extended by Necir et al., (Journal of Probability and Statistics 596839:17 2010) in the case of infinite second moment. In this framework, we propose a reduced-bias estimator of the CTE. We illustrate the efficiency of our approach on a small simulation study and a real data analysis.



The authors thank the referee for the comments concerning the previous version. The first author acknowledges support from AIRES-Sud (AIRES-Sud is a program from the French Ministry of Foreign and European Affairs, implemented by the “Institut de Recherche pour le Développement”, IRD-DSF) and from the “Ministère de la Recherche Scientifique” of Sénégal.


  1. Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9, 203–228.CrossRefzbMATHMathSciNetGoogle Scholar
  2. Beirlant, J., Dierckx, G., Goegebeur, M., & Matthys, G. (1999). Tail index estimation and an exponential regression model. Extremes, 2, 177–200.CrossRefzbMATHMathSciNetGoogle Scholar
  3. Beirlant, J., Dierckx, G., Guillou, A., & Starica, C. (2002). On exponential representations of log-spacings of extreme order statistics. Extremes, 5, 157–180.CrossRefzbMATHMathSciNetGoogle Scholar
  4. Bingham, N. H., Goldie, C. M., & Teugels, J. L. (1987). Regular variation, Cambridge.Google Scholar
  5. Brazauskas, V., Jones, B., Puri, M., & Zitikis, R. (2008). Estimating conditional tail expectation with actuarial applications in view. Journal of Statistical Planning and Inference, 138, 3590–3604.CrossRefzbMATHMathSciNetGoogle Scholar
  6. Csörgő, M., Csörgő, S., Horváth, L., & Mason, D. M. (1986). Weighted empirical and quantile processes. Annals of Probability, 14, 31–85.CrossRefMathSciNetGoogle Scholar
  7. Csörgő, S., Deheuvels, P., & Mason, D. M. (1985). Kernel estimates of the tail index of a distribution. Annals of Statistics, 13, 1050–1077.CrossRefMathSciNetGoogle Scholar
  8. de Haan, L., & Ferreira, A. (2006). Extreme value theory: An introduction. Springer.Google Scholar
  9. Deme, E., Gardes, L., & Girard, S. (2013a). On the estimation of the second order parameter for heavy-tailed distributions. REVSTAT—Statistical Journal, 11(3), 277–299.Google Scholar
  10. Deme, E., Girard, S., & Guillou, A. (2013b). Reduced-bias estimator of the proportional hazard premium for heavy-tailed distributions. Insurance Mathematic & Economics, 52, 550–559.CrossRefzbMATHMathSciNetGoogle Scholar
  11. Feuerverger, A., & Hall, P. (1999). Estimating a tail exponent by modelling departure from a Pareto distribution. Annals of Statistics, 27, 760–781.CrossRefzbMATHMathSciNetGoogle Scholar
  12. Fraga Alves, M. I., Gomes, M. I., & de Haan, L. (2003). A new class of semi-parametric estimators of the second order parameter. Portugaliae Mathematica, 60(2), 193–213.Google Scholar
  13. Geluk, J. L., & de Haan, L. (1987). Regular variation, extensions and Tauberian theorems, CWI tract 40, Center for Mathematics and Computer Science, P.O. Box 4079, 1009 AB Amsterdam, The Netherlands.Google Scholar
  14. Goovaerts, M. J., de Vylder, F., & Haezendonck, J. (1984). Insurance premiums, theory and applications. Amsterdam: North Holland.Google Scholar
  15. Hill, B. M. (1975). A simple approach to inference about the tail of a distribution. Annals of Statistics, 3, 1136–1174.CrossRefGoogle Scholar
  16. Landsman, Z., & Valdez, E. (2003). Tail conditional expectations for elliptical distributions. North American Actuarial Journal, 7, 55–71.CrossRefzbMATHMathSciNetGoogle Scholar
  17. Matthys, G., Delafosse, E., Guillou, A., & Beirlant, J. (2004). Estimating catastrophic quantile levels for heavy-tailed distributions. Insurance Mathematic & Economics, 34, 517–537.CrossRefzbMATHMathSciNetGoogle Scholar
  18. McNeil, A.J., Frey, R., & Embrechts, P. (2005) Quantitative risk management: concepts, techniques, and tools, Princeton University Press.Google Scholar
  19. Necir, A., Rassoul, A., & Zitikis, R. (2010) Estimating the conditional tail expectation in the case of heavy-tailed losses, Journal of Probability and Statistics, ID 596839, 17.Google Scholar
  20. Pan, X., Leng, X., & Hu, T. (2013). Second-order version of Karamata’s theorem with applications. Statistics and Probability Letters, 83, 1397–1403.CrossRefzbMATHMathSciNetGoogle Scholar
  21. Weissman, I. (1978). Estimation of parameters and larges quantiles based on the \(k\) largest observations. Journal of American Statistical Association, 73, 812–815.zbMATHMathSciNetGoogle Scholar
  22. Zhu, L., & Li, H. (2012). Asymptotic analysis of multivariate tail conditional expectations. North American Actuarial Journal, 16, 350–363.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • El Hadji Deme
    • 1
  • Stéphane Girard
    • 2
  • Armelle Guillou
    • 3
    Email author
  1. 1.LERSTADUniversité Gaston Berger de Saint-LouisSaint-LouisSénégal
  2. 2.Team MistisInria Grenoble Rhône-Alpes & Laboratoire Jean KuntzmannMontbonnotFrance
  3. 3.Université de Strasbourg & CNRS, IRMA, UMR 7501StrasbourgFrance

Personalised recommendations