Estimation of the causal dose-response curve is an old problem in statistics. In a non parametric model, if the treatment is continuous, the dose-response curve is not a pathwise differentiable parameter, and no root-n-consistent estimator is available. However, the risk of a candidate algorithm for estimation of the dose response curve is a pathwise differentiable parameter, whose consistent and efficient estimation is possible. In this work, we review the cross validated augmented inverse probability of treatment weighted estimator (CV A-IPTW) of the risk, and present a cross validated targeted minimum loss based estimator (CV-TMLE) counterpart. These estimators are proven consistent an efficient under certain consistency and regularity conditions on the initial estimators of the outcome and treatment mechanism. We also present a methodology that uses these estimated risks to select among a library of candidate algorithms. These selectors are proven optimal in the sense that they are asymptotically equivalent to the oracle selector under certain consistency conditions on the estimators of the treatment and outcome mechanisms. Because the CV-TMLE is a substitution estimator, it is more robust than the CV-AIPTW against empirical violations of the positivity assumption. This and other small sample size differences between the CV-TMLE and the CV-A-IPTW are explored in a simulation study.
Díaz, Iván and van der Laan, Mark J., "Targeted Data Adaptive Estimation of the Causal Dose Response Curve" (January 2013). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 306.