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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: [email protected]
Kai Wang Ng
Affiliation:
Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong E-mail: [email protected]
Taizhong Hu
Affiliation:
Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, China E-mail: [email protected]
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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 

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