Asymptotic Normality of the Conditional Value-at-Risk based Pickands Estimator

Abstract: 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.