Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9, 203–228.
Beirlant, J., Dierckx, G., Goegebeur, M., & Matthys, G. (1999). Tail index estimation and an exponential regression model. Extremes, 2, 177–200.
Beirlant, J., Dierckx, G., Guillou, A., & Starica, C. (2002). On exponential representations of log-spacings of extreme order statistics. Extremes, 5, 157–180.
Bingham, N. H., Goldie, C. M., & Teugels, J. L. (1987). Regular variation, Cambridge.
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.
Csörgő, M., Csörgő, S., Horváth, L., & Mason, D. M. (1986). Weighted empirical and quantile processes. Annals of Probability, 14, 31–85.
Csörgő, S., Deheuvels, P., & Mason, D. M. (1985). Kernel estimates of the tail index of a distribution. Annals of Statistics, 13, 1050–1077.
de Haan, L., & Ferreira, A. (2006). Extreme value theory: An introduction. Springer.
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.
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.
Feuerverger, A., & Hall, P. (1999). Estimating a tail exponent by modelling departure from a Pareto distribution. Annals of Statistics, 27, 760–781.
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.
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.
Goovaerts, M. J., de Vylder, F., & Haezendonck, J. (1984). Insurance premiums, theory and applications. Amsterdam: North Holland.
Hill, B. M. (1975). A simple approach to inference about the tail of a distribution. Annals of Statistics, 3, 1136–1174.
Landsman, Z., & Valdez, E. (2003). Tail conditional expectations for elliptical distributions. North American Actuarial Journal, 7, 55–71.
Matthys, G., Delafosse, E., Guillou, A., & Beirlant, J. (2004). Estimating catastrophic quantile levels for heavy-tailed distributions. Insurance Mathematic & Economics, 34, 517–537.
McNeil, A.J., Frey, R., & Embrechts, P. (2005) Quantitative risk management: concepts, techniques, and tools, Princeton University Press.
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.
Pan, X., Leng, X., & Hu, T. (2013). Second-order version of Karamata’s theorem with applications. Statistics and Probability Letters, 83, 1397–1403.
Weissman, I. (1978). Estimation of parameters and larges quantiles based on the \(k\) largest observations. Journal of American Statistical Association, 73, 812–815.
Zhu, L., & Li, H. (2012). Asymptotic analysis of multivariate tail conditional expectations. North American Actuarial Journal, 16, 350–363.
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Deme, E.H., Girard, S., Guillou, A. (2015). Reduced-Bias Estimator of the Conditional Tail Expectation of Heavy-Tailed Distributions. In: Hallin, M., Mason, D., Pfeifer, D., Steinebach, J. (eds) Mathematical Statistics and Limit Theorems. Springer, Cham. https://doi.org/10.1007/978-3-319-12442-1_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-12442-1_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-12441-4
Online ISBN: 978-3-319-12442-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)