Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-23T11:10:18.735Z Has data issue: false hasContentIssue false
  • English
  • Français

ASYMPTOTIC EXPANSIONS OF GENERALIZED QUANTILES AND EXPECTILES FOR EXTREME RISKS

Published online by Cambridge University Press:  16 April 2015

Tiantian Mao ,
Kai Wang Ng  and
Taizhong Hu
Tiantian Mao
Affiliation:
Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, China E-mail: tmao@ustc.edu.cn
Kai Wang Ng
Affiliation:
Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong E-mail: kaing@hku.hk
Taizhong Hu
Affiliation:
Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, China E-mail: thu@ustc.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

Generalized quantiles of a random variable were defined as the minimizers of a general asymmetric loss function, which include quantiles, expectiles and M-quantiles as their special cases. Expectiles have been suggested as potentially better alternatives to both Value-at-Risk and expected shortfall risk measures. In this paper, we first establish the first-order expansions of generalized quantiles for extreme risks as the confidence level α↑ 1, and then investigate the first-order and/or second-order expansions of expectiles of an extreme risk as α↑ 1 according to the underlying distribution belonging to the max-domain of attraction of the Fréchet, Weibull, and Gumbel distributions, respectively. Examples are also presented to examine whether and how much the first-order expansions have been improved by the second-order expansions.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 29 , Issue 3 , July 2015 , pp. 309 - 327
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Artzner, P., Delbaen, F., Eber, J.-M. & Heath, D. (1999). Coherent measures of risk. Mathematical Finance 9: 203–228.Google Scholar
2. Bellini, F. & Bignozzi, V. (forthcoming). Elicitable risk measures. Quantitative Finance doi:10.1080/14697688.2014.946955.Google Scholar
3. Bellini, F., Klar, B., Müller, A. & Gianin, E.R. (2014). Generalized quantiles as risk measures. Insurance: Mathematics and Economics 54: 41–48.Google Scholar
4. Bingham, N.H., Goldie, C.M. & Teugels, J.L. (1987). Regular variation. Cambridge: Cambridge University Press.Google Scholar
5. Breckling, J. & Chambers, R. (1988). M-quantiles. Biometrika 75: 761–772.Google Scholar
6. Chen, Z. (1996). Conditional L p -quantiles and their application to the testing of symmetry in non-parametric regression. Statistics and Probability Letters 29: 107–115.Google Scholar
7. de Haan, L. & Ferreira, A. (2006). Extreme value theory: an introduction. Springer Series in Operations Research and Financial Engineering. New York: Springer.Google Scholar
8. De Rossi, G. & Harvey, A. (2009). Quantiles, expectiles and splines. Journal of Econometrics 152: 179–185.Google Scholar
9. Delbaen, F. (2013). A remark on the structure of expectiles. Preprint no. arXiv:1307.5881. ETH Zurich.Google Scholar
10. Denuit, M., Dhaene, J., Goovaerts, M.J. & Kaas, R. (2005). Actuarial theory for dependent risks: measures, orders and models. West Sussex: John Wiley & Sons, Ltd.Google Scholar
11. Embrechts, P., Klüppelberg, C. & Mikosch, T. (1997). Modelling extremal events for finance and insurance. Berlin: Springer-Verlag.Google Scholar
12. Embrechts, P., Puccetti, G., Rüschendorf, L., Wang, R. & Beleraj, A. (2013). An academic response to Basel 3.5. Risks 2: 25–48.Google Scholar
13. Emmer, S., Kratz, M. & Tasche, D. (2013). What is the best risk measure in practice? A comparison of standard measures. ArXiv:1312.1645v2.Google Scholar
14. Galambos, J. (2001). The asymptotic theory of extreme order statistics. 2nd ed. New York: Robert E. Krieger Publishing Co., Inc.Google Scholar
15. Gneiting, T. (2011). Making and evaluating point forecasts. Journal of the American Statistical Association 106: 746–762.Google Scholar
16. Hua, L. & Joe, H. (2011). Second order regular variation and conditional tail expectation of multiple risks. Insurance: Mathematics and Economics 49: 537–546.Google Scholar
17. Koenker, R. (2005). Quantile Regression. New York: Cambridge University Press.Google Scholar
18. Kuan, C.-M., Yeh, J.-H. & Hsu, Y.-C. (2009). Assessing value at risk with CARE, the conditional autoregressive expectile models. Journal of Econometrics 150: 261–270.Google Scholar
19. Lv, W., Mao, T. & Hu, T. (2012). Properties of second-order regular variation and expansions for risk concentration. Probability in the Engineering and Informational Sciences 26: 535–559.Google Scholar
20. Mao, T. & Hu, T. (2012). Second-order properties of the Haezendonck–Goovaerts risk measure for extreme risks. Insurance: Mathematics and Economics 51: 333–343.Google Scholar
21. Mao, T. & Hu, T. (2013). Second-order properties of risk concentrations without the condition of asymptotic smoothness. Extremes 16: 383–405.Google Scholar
22. Newey, W. & Powell, J. (1987). Asymptotic least square estimation and testing. Econometrica 55: 819–847.Google Scholar
23. Resnick, S.I. (2007). Heavy-tail phenomena. Springer Series in Operations Research and Financial Engineering. New York: Springer.Google Scholar
24. Ziegel, J.F. (2014). Coherence and elicitability. Mathematical Finance, doi: 10.1111/mafi.12080.Google Scholar