We consider a targeted maximum likelihood estimator of a path-wise differentiable parameter of the data generating distribution in a semi-parametric model based on observing n independent and identically distributed observations. The targeted maximum likelihood estimator (TMLE) uses V-fold sample splitting for the initial estimator in order to make the TMLE maximally robust in its bias reduction step. We prove a general theorem that states asymptotic efficiency (and thereby regularity) of the targeted maximum likelihood estimator when the initial estimator is consistent and a second order term converges to zero in probability at a rate faster than the square root of the sample size, but no other meaningful conditions are needed. In particular, the conditions of this theorem allow the full utilization of loss based super learning to obtain the initial estimator.
In particular, the theorem proves that first order efficient and unbiased estimation is enhanced in an important way by using adaptive estimators such as an super learner, thereby formally dealing with the concern that adaptive estimation might make it harder to construct valid confidence intervals. On the contrary, the theorem teaches us that to achieve first order efficiency and regularity, it is crucial to estimate the relevant parts of the true data generating distribution as good as possible. The theorem is applied to prove asymptotic efficiency of the targeted maximum likelihood estimator of the additive causal effect of a binary treatment on an outcome in a randomized controlled trial and in an observational study. Excellent finite sample performance of this estimator has been demonstrated in past articles (e.g.van der Laan et al. (September, 2009), Gruber and van der Laan (2010), Stitelman and van der Laan (2010), Petersen et al. (2010).
Zheng, Wenjing and van der Laan, Mark J., "Asymptotic Theory for Cross-validated Targeted Maximum Likelihood Estimation" (November 2010). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 273.