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

Abstract

The tail distortion risk measure at level p was first introduced in Zhu and Li (2012), 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.

Keywords:

• Asymptotics
• Extended regular variation
• Max-domain of attraction
• Regular variation
• Second-order condition
• Tail distortion risk measure

Mathemetics Subject Classification:

• Primary 62P05
• Secondary 60G70, 62E20