Inverting estimating equations for causal inference on quantiles

The causal inference literature frequently focuses on estimating the mean of the potential outcome, whereas quantiles of the potential outcome may carry important additional information. We propose an inverse estimating equation framework to generalize a wide class of causal inference solutions from estimating the mean of the potential outcome to its quantiles. We assume that a moment function is available to identify the mean of the threshold-transformed potential outcome, based on which a convenient construction of the estimating equation of the quantiles of the potential outcome is proposed. In addition, we give a general construction of the efficient influence functions of the mean and quantiles of potential outcomes, and establish their connection. We motivate estimators for the quantile estimands with the efficient influence function, and develop their asymptotic properties when either parametric models or data-adaptive machine learners are used to estimate the nuisance functions. A broad implication of our results is that one can rework the existing result for mean causal estimands to facilitate causal inference on quantiles. Our general results are illustrated by several analytical and numerical examples.