Statistical inference for tail-based cumulative residual entropy

Tail risk analysis plays a central strategic role in risk management, and focuses on the problem of risk measurement in the tail regions of extreme risks. In this paper, by extending the entropy method, we propose a quantity called Tail-based Cumulative Residual Entropy (TCRE) in a bivariate set-up as an alternative measurement of qualifying tail risk variability, which is less addressed in the literature. The TCRE in a bivariate set-up can be viewed as the tail cumulative residual entropy for a random vector (X,Y) with some dependence structure, and the combination of it with the Marginal Expected Shortfall (MES) constitutes a new risk measure called Marginal CRE Shortfall (MCRES). For a wide class of bivariate extreme value distributions, we construct a nonparametric estimator of the proposed TCRE measure and establish an asymptotic distribution of the estimator with extreme high-level tail. We use extreme value techniques to calculate the limit of the proposed measure as the risk level approaches to the extreme status and present how the tail dependence structure and marginal risk severity have an influence on its limits. Furthermore, a simulation study and real data analysis of the proposed TCRE are presented to understand and verify the properties of TCRE and its estimators.