Asymptotically effcient estimation of the conditionalexpected shortfall

We propose a procedure for efficient estimation of the trimmed mean of a random variable Y conditional on a set of covariates X. For concreteness, we focus on a financial application where the trimmed mean of interest corresponds to a coherent measure of risk, namely the conditional expected shor… tfall. Our estimator is based on the representation of the estimand as an integral of the conditional quantile function. We extend the class of estimators originally proposed by Peracchi and Tanase (2008) by introducing a weighting function that gives different weights to different conditional quantiles. Our approach allows for either parametric or nonparametric modeling of the conditional quantiles and the weights, but is essentially nonparametric in spirit. We prove consistency and asymptotic normality of the resulting estimator. Optimizing over the weighting function, we obtain asymptotic efficiency gains with respect to the unweighted estimators. The gains are especially noticeable in the case of fat-tailed distributions.