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Efficient Estimation of Quantiles in Missing Data Models

Ivan Diaz
Google Inc., 111 8th Avenue (2015)


We present a method to estimate the quantile of a variable subject to missingness, under the missing at random assumption. Our proposed estimator is locally efficient, root-n-consistent, asymptotically normal, and doubly robust, under regularity conditions. We use Monte Carlo simulation to compare our proposal to the one-step and inverse-probability weighted estimators. Our estimator is superior to both competitors, with a mean squared error up to 8 times smaller than the one-step estimator, and up to 2.5 times smaller than an inverse probability weighted estimator. We develop extensions for estimating the causal effect of treatment on a population quantile among the treated. Our methods are motivated by an application with a heavy tailed continuous outcome. In this situation, the efficiency bound for estimating the effect on the mean is often large or infinite, ruling out root-n-consistent inference and reducing the power for testing hypothesis of no treatment effect. Using quantiles (e.g., the median) may yield more accurate measures of the treatment effect, along with more powerful hypothesis tests. In our application, the proposed estimator of the effect on the median yields hypothesis tests of no treatment effect up to two times more powerful, and its variance is up to four times smaller than the variance of its mean counterpart.