First- and Second-order Asymptotics for the Tail Distortion Risk Measure of Extreme Risks

The tail distortion risk measure at level p was first introduced in Zhu and Li (2012 Zhu, L., Li, H. (2012). Tail distortion risk and its asymptotic analysis. Insur. Math. Econ. 51(1):115–121.[Crossref], [Web of Science ®] , [Google Scholar]), where the parameter p ∈ (0, 1) indicates the confidence level. They established first-order asymptotics for this risk measure, as p↑1, for the Fréchet case. In this article, we extend their work by establishing both first-order and second-order asymptotics for the Fréchet, Weibull, and Gumbel cases. Numerical studies are also carried out to examine the accuracy of both asymptotics.