Asymptotic Properties of Empirical Quantile-Based Estimators

We consider the problem of performing inference on the mathematical expectation of unknown quantile-cdf transforms of a random variable. A prominent instance where this problem arises is the Changes-in-Changes causal inference model developed by Athey and Imbens (2006), in which the average treatment effect takes this form. We propose a new inference procedure and asymptotic theory based on a simple plug-in estimator, avoiding the need for differentiability assumptions, bounded densities, and the functional delta method. We establish asymptotic normality at the parametric root-n rate under weak moment and tail conditions. We also develop a new estimator of the asymptotic variance and show its consistency under Hölder-type smoothness conditions on the densities. Our analysis leverages existing results from the theory of L-statistics and new results on the empirical process that may be of independent interest. Taken together, these results offer improved robustness and broaden the scope of application of nonlinear methods for causal inference.