In order to be concrete we focus on estimation of the treatment specific mean, controlling for all measured baseline covariates, based on observing n independent and identically distributed copies of a random variable consisting of baseline covariates, a subsequently assigned binary treatment, and a final outcome. The statistical model only assumes possible restrictions on the conditional distribution of treatment, given the covariates, the so called propensity score. Estimators of the treatment specific mean involve estimation of the propensity score and/or estimation of the conditional mean of the outcome, given the treatment and covariates. In order to make these estimators asymptotically unbiased at any data distribution in the statistical model, it is essential to use data adaptive estimators of these nuisance parameters such as ensemble learning, and specifically super-learning. Because such estimators involve optimal trade-off of bias and variance w.r.t. the infinite dimensional nuisance parameter itself, they result in a sub-optimal bias/variance trade-off for the resulting real valued estimator of the estimand. We demonstrate that additional targeting of the estimators of these nuisance parameters guarantees that this bias for the estimand is second order, and thereby allows us to prove theorems thatestablish asymptotic linearity of the estimator of the treatment specific mean under regularity conditions. These insights result in novel targeted maximum likelihood estimators (TMLE) that use ensemble learning withadditional targeted bias reduction to construct estimators of the nuisance parameters. In particular, we construct collaborative targeted maximum likelihood estimators (CTMLE) with known influence curve allowing for statistical inference, even though these CTMLEs involve variable selection for the propensity score based on a criterion that measures how effective the resulting fit of the propensity score is in removing bias for the estimand. As a particular special case, we also demonstrate the required targeting of the propensity score for the inverse probability of treatment weighted estimator using super-learning to fit the propensity score.
van der Laan, Mark J., "Statistical Inference when using Data Adaptive Estimators of Nuisance Parameters" (November 2012). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 302.