Asymptotic Behaviors of the VaR and CVaR Estimates for Widely Orthant Dependent Sequences

This paper considers some asymptotics of value-at-risk (VaR) and conditional value-at-risk (CVaR) estimates in the cases of extended negatively dependent (END) and widely orthant dependent (WOD) sequences. The Bahadur representation and strong consistency of VaR estimator are obtained, and the strong convergence rate of CVaR estimator is obtained based on END and WOD sequences. In addition, the asymptotic normality of VaR estimator is given based on END sequence. Finally, some simulations to study the numerical performance of the consistency for VaR and CVaR estimators are given in the case of WOD sample. Our results extend and improve some corresponding results.