Mark van der Laan: Higher order Targeted Maximum Likelihood Estimation

"Higher order Targeted Maximum Likelihood Estimation"
Mark van der Laan, UC Berkeley
Discussant: Alex Luedtke, University of Washington

Abstract: Asymptotic linearity and efficiency of targeted maximum likelihood estimators (TMLE) of target features of the data distribution relies on a a second order remainder being asymptotically negligible. 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 ATE, and of the integrated squared density.

January 19, 2020