Consider a study in which one observes n independent and identically distributed random variables whose probability distribution is known to be an element of a particular statistical model, and one is concerned with estimation of a particular real valued pathwise differentiable target parameter of this data probability distribution. The canonical gradient of the pathwise derivative of the target parameter, also called the efficient influence curve, defines an asymptotically efficient estimator as an estimator that is asymptotically linear with influence curve equal to the efficient influence curve.The targeted maximum likelihood estimator is a two stage estimator obtained by constructing a so called least favorable parametric submodel through an initial estimator with score, at zero fluctuation of the initial estimator, that spans the efficient influence curve, and iteratively maximizing the corresponding parametric likelihood till no more updates occur, at which point the updated initial estimator solves the so called efficient influence curve equation. The latter property establishes the asymptotic efficiency of the TMLE under appropriate conditions, including that the initial estimator is within a neighborhood of the true data distribution.
In this article we construct a one-dimensional universal least favorable submodel for which the TMLE only takes one step, and thereby requires minimal extra fitting with data to achieve its goal of solving the efficient influence curve equation. We generalize these to universal least favorable submodels through the relevant part of the data distribution as required for targeted minimum loss-based estimation, and to universal score-specific submodels for solving any other desired equation beyond the efficient influence curve equation. We demonstrate the one-step targeted minimum loss-based estimators based on such universal least favorable submodels for a variety of examples showing that any of the goals for TMLE we previously achieved with local (typically multivariate) least favorable parametric submodels and an iterative TMLE can also be achieved with our new one-dimensional universal least favorable submodels, resulting in new one-step TMLEs for a large class of estimation problems previously addressed. Finally, remarkably, given a multidimensional target parameter, we develop a universal canonical one-dimensional submodel such that the one-step TMLE, only maximizing the log-likelihood over a univariate parameter, solves the multivariate efficient influence curve equation. This allows us to construct a one-step TMLE based on a one-dimensional parametric submodel through the initial estimator, that solves any multivariate desired set of estimating equations.
van der Laan, Mark J., "One-Step Targeted Minimum Loss-based Estimation Based on Universal Least Favorable One-Dimensional Submodels" (June 2015). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 338.