Asymptotic Theory for Cross-validated Targeted Maximum Likelihood Estimation

Wenjing Zheng, University of California, Berkeley, Division of Biostatistics
Mark J. van der Laan, University of California, Berkeley

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This material is published in: W. Zheng, M.J. van der Laan (2011). "Cross-Validated Targeted Minimum-Loss-Based Estimation." In M.J. van der Laan and S. Rose, Targeted Learning: Causal Inference for Observational and Experimental Data, Chapter 27. New York, Springer.

Abstract

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).