  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.
