Title: Higher-order targeted maximum likelihood estimation and its applications in causal inference
Authors: Mark van der Laan - University of California Berkeley (United States) [presenting]
Abstract: Asymptotic linearity and efficiency of targeted maximum likelihood estimators (TMLE) of target features of the data distribution rely on a second-order remainder being asymptotically negligible. The Highly Adaptive Lasso (HAL) provides a general nonparametric MLE that controls this remainder at the desired level, only assuming that the target functional parameters are cadlag and have finite sectional variation norm. However, in finite samples, the second-order remainder can dominate the sampling distribution so that inference based on asymptotic normality would be anti-conservative. We propose a new higher-order (say $k$-th order) TMLE, generalizing the regular (first-order) TMLE. We prove that it satisfies an exact linear expansion, in terms of efficient influence functions of sequentially defined higher-order fluctuations of the target parameter, with a remainder that is a $k+1$-th order remainder. As a consequence, this $k$-th order TMLE allows statistical inference only relying on the $k+1$-th order remainder being negligible. We present the theoretical result as well as simulations for the second-order TMLE for nonparametric estimation of the causal quantities, and of the integrated squared density. Its general applications in causal inference for optimal adjustment for baseline and time-dependent confounders is highlighted as well. We also discuss advances in computing higher-order efficient influence functions utilizing HAL.