Nonparametric estimation of expected shortfall via Bahadur-type representation and Berry–Esséen bounds

The expected shortfall is an important risk measure in financial risk management. In this paper, we study the Bahadur-type representation of an improved nonparametric expected shortfall estimator for φ-mixing financial losses without any restrictions on the mixing rates. The result established in this work improves and extends some existing ones in the literature. Based on the Bahadur-type representation, we further establish the Berry–Esséen bound for the modified nonparametric expected shortfall estimator. We show that the optimal rate can achieve nearly under some suitable conditions. We also carry out some numerical simulations and a real data analysis to support the theoretical results established here.