Nonparametric estimation of expected shortfall for α-mixing financial losses.

In this paper, we investigate the Bahadur type representation of a nonparametric expected shortfall estimator for α -mixing financial losses. Based on the Bahadur type representation, we further establish the Berry–Esseen bound of the nonparametric expected shortfall estimator. It is shown that the optimal rate can achieve nearly O (n - 1 / 8) under some appropriate conditions. We also carry out some numerical simulations and a real data example to support our theoretical results based on finite samples. [ABSTRACT FROM AUTHOR]