Nonparametric inference for distortion risk measures on tail regions

Suppose X is some interesting loss and Y is a benchmark variable. Given some extreme scenarios of Y, it is indispensable to measure the tail risk of X by applying a class of univariate risk measures to study the co-movement of the two variables. In this paper, we consider the extreme and nonparametric inference for the distortion risk measures on the tail regions when the extreme scenarios of some benchmark variable are considered. We derive the limit of the proposed risk measures based on Extreme Value Theory. The asymptotics of the risk measures shows the decomposition of the marginal extreme value index and the extreme dependence structure which implies how these two pieces of information have influences on the limit of the risk measures. Finally, for practical purpose, we develop a nonparametric estimation method for the distortion risk measures on tail regions and its asymptotic normality is derived. (C) 2019 Elsevier B.V. All rights reserved.