Efficient Estimation of Quantiles in Missing Data Models

Abstract: We propose a novel targeted maximum likelihood estimator (TMLE) for quantiles in semiparametric missing data models. Our proposed estimator is locally efficient, $\sqrt{n}$-consistent, asymptotically normal, and doubly robust, under regularity conditions. We use Monte Carlo simulation to compare our proposed method to existing estimators. The TMLE is superior to all competitors, with relative efficiency up to three times smaller than the inverse probability weighted estimator (IPW), and up to two times smaller than the augmented IPW. This research is motivated by a causal inference research question with highly variable treatment assignment probabilities, and a heavy tailed, highly variable outcome. Estimation of causal effects on the mean is a hard problem in such scenarios because the information bound is generally small. In our application, the efficiency bound for estimating the effect on the mean is possibly infinite. This rules out $\sqrt{n}$-consistent inference and reduces the power for testing hypothesis of no treatment effect on the mean. In our simulations, using the effect on the median allows us to test a location-shift hypothesis with 30\% more power. This allows us to make claims about the effectiveness of treatment that would have hard to make for the effect on the mean. We provide R code to implement the proposed estimators.