Skip to main content Accessibility help
×
Home
Hostname: page-component-747cfc64b6-db5sh Total loading time: 0.258 Render date: 2021-06-13T21:42:58.940Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

ASYMPTOTIC EXPANSIONS OF GENERALIZED QUANTILES AND EXPECTILES FOR EXTREME RISKS

Published online by Cambridge University Press:  16 April 2015

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
Corresponding

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
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.

References

1. Artzner, P., Delbaen, F., Eber, J.-M. & Heath, D. (1999). Coherent measures of risk. Mathematical Finance 9: 203228.CrossRefGoogle 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: 4148.Google Scholar
4. Bingham, N.H., Goldie, C.M. & Teugels, J.L. (1987). Regular variation. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
5. Breckling, J. & Chambers, R. (1988). M-quantiles. Biometrika 75: 761772.CrossRefGoogle 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: 107115.CrossRefGoogle Scholar
7. de Haan, L. & Ferreira, A. (2006). Extreme value theory: an introduction. Springer Series in Operations Research and Financial Engineering. New York: Springer.CrossRefGoogle Scholar
8. De Rossi, G. & Harvey, A. (2009). Quantiles, expectiles and splines. Journal of Econometrics 152: 179185.CrossRefGoogle 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.CrossRefGoogle Scholar
11. Embrechts, P., Klüppelberg, C. & Mikosch, T. (1997). Modelling extremal events for finance and insurance. Berlin: Springer-Verlag.CrossRefGoogle Scholar
12. Embrechts, P., Puccetti, G., Rüschendorf, L., Wang, R. & Beleraj, A. (2013). An academic response to Basel 3.5. Risks 2: 2548.CrossRefGoogle 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: 746762.CrossRefGoogle Scholar
16. Hua, L. & Joe, H. (2011). Second order regular variation and conditional tail expectation of multiple risks. Insurance: Mathematics and Economics 49: 537546.Google Scholar
17. Koenker, R. (2005). Quantile Regression. New York: Cambridge University Press.CrossRefGoogle 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: 261270.CrossRefGoogle 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: 535559.CrossRefGoogle Scholar
20. Mao, T. & Hu, T. (2012). Second-order properties of the Haezendonck–Goovaerts risk measure for extreme risks. Insurance: Mathematics and Economics 51: 333343.Google Scholar
21. Mao, T. & Hu, T. (2013). Second-order properties of risk concentrations without the condition of asymptotic smoothness. Extremes 16: 383405.CrossRefGoogle Scholar
22. Newey, W. & Powell, J. (1987). Asymptotic least square estimation and testing. Econometrica 55: 819847.CrossRefGoogle 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
13
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

ASYMPTOTIC EXPANSIONS OF GENERALIZED QUANTILES AND EXPECTILES FOR EXTREME RISKS
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

ASYMPTOTIC EXPANSIONS OF GENERALIZED QUANTILES AND EXPECTILES FOR EXTREME RISKS
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

ASYMPTOTIC EXPANSIONS OF GENERALIZED QUANTILES AND EXPECTILES FOR EXTREME RISKS
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *