Asymptotically Efficient Estimation of the Conditional Expected Shortfall

This paper proposes a procedure for efficient estimation of the trimmed mean of a random variable conditional on a set of covariates. For concreteness, the paper focuses on a financial application where the trimmed mean of interest corresponds to the conditional expected shortfall, which is known to be a coherent risk measure. The proposed class of estimators is based on representing the estimand as an integral of the conditional quantile function. Relative to the simple analog estimator that weights all conditional quantiles equally, asymptotic efficiency gains may be attained by giving different weights to the different conditional quantiles while penalizing excessive departures from uniform weighting. The approach presented here allows for either parametric or nonparametric modeling of the conditional quantiles and the weights, but is essentially nonparametric in spirit. The paper establishes the asymptotic properties of the proposed class of estimators. Their finite sample properties are illustrated through a set of Monte Carlo experiments and an empirical application.