Common approaches to parametric statistical inference often encounter difficulties in the context of infinite-dimensional models. The framework of targeted maximum likelihood estimation (TMLE), introduced in van der Laan & Rubin (2006), is a principled approach for constructing asymptotically linear and efficient substitution estimators in rich infinite-dimensional models. The mechanics of TMLE hinge upon first-order approximations of the parameter of interest as a mapping on the space of probability distributions. For such approximations to hold, a second-order remainder term must tend to zero sufficiently fast. In practice, this means an initial estimator of the underlying data-generating distribution with a sufficiently large rate of convergence must be available -- in many cases, this requirement is prohibitively difficult to satisfy. In this article, we propose a generalization of TMLE utilizing a higher-order approximation of the target parameter. This approach yields asymptotically linear and efficient estimators when a higher-order remainder term is asymptotically negligible. The latter condition is often much less stringent than that arising in a regular first-order TMLE. Beyond relaxing regularity conditions, use of a higher-order TMLE can improve inference accuracy in finite samples due to its explicit reliance on a higher-order approximation. We provide the theoretical foundations of higher-order TMLE and study its use for estimating a counterfactual mean when all potential confounders have been measured. We show, in particular, that the implementation of a higher-order TMLE is nearly identical to that of a regular first-order TMLE. Since higher-order TMLE requires higher-order differentiability of the target parameter, a requirement that often fails to hold, we also discuss and study practicable approximation strategies that allow us to circumvent this failure in applications.
Statistical Methodology | Statistical Theory
Carone, Marco; Díaz, Iván; and van der Laan, Mark J., "Higher-order Targeted Minimum Loss-based Estimation" (December 2014). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 331.