The Pickands estimator for the extreme value index is beneficial due to its universal consistency, location, and scale invariance, which sets it apart from other types of estimators. However, similar to many extreme value index estimators, it is marked by poor asymptotic efficiency. Chen (2021) introduces a Conditional Value-at-Risk (CVaR)-based Pickands estimator, establishes its consistency, and demonstrates through simulations that this estimator significantly reduces mean squared error while preserving its location and scale invariance. The initial focus of this paper is on demonstrating the weak convergence of the empirical CVaR in functional space. Subsequently, based on the established weak convergence, the paper presents the asymptotic normality of the CVaR-based Pickands estimator. It further supports these theoretical findings with empirical evidence obtained through simulation studies.

翻译：Pickands估计量因其普适一致性、位置与尺度不变性而在极值指数估计中具有独特优势，这使其区别于其他类型的估计量。然而，与许多极值指数估计量类似，该估计量存在渐近效率较低的缺陷。Chen (2021) 提出了一种基于条件风险价值（CVaR）的Pickands估计量，证明了其一致性，并通过模拟实验表明该估计量在保持位置与尺度不变性的同时，能显著降低均方误差。本文首先证明经验CVaR在函数空间中的弱收敛性。随后，基于已建立的弱收敛结果，推导出基于CVaR的Pickands估计量的渐近正态性。最后，通过模拟研究获得的实证证据进一步支持了上述理论结论。