Targeted Maximum Likelihood Estimation using Exponential Families

TL;DR

This paper introduces a novel approach to targeted maximum likelihood estimation (TMLE) using exponential families, enhancing computational efficiency and robustness in estimating parameters in complex models.

Contribution

It proposes using exponential families for TMLE submodels, providing a general, computationally efficient method applicable to various smooth parameters in nonparametric models.

Findings

Exponential family submodels improve computational efficiency.

Choice of submodel significantly affects finite-sample performance.

The method is demonstrated in three diverse estimation problems.

Abstract

Targeted maximum likelihood estimation (TMLE) is a general method for estimating parameters in semiparametric and nonparametric models. Each iteration of TMLE involves fitting a parametric submodel that targets the parameter of interest. We investigate the use of exponential families to define the parametric submodel. This implementation of TMLE gives a general approach for estimating any smooth parameter in the nonparametric model. A computational advantage of this approach is that each iteration of TMLE involves estimation of a parameter in an exponential family, which is a convex optimization problem for which software implementing reliable and computationally efficient methods exists. We illustrate the method in three estimation problems, involving the mean of an outcome missing at random, the parameter of a median regression model, and the causal effect of a continuous exposure, respectively. We conduct a simulation study comparing different choices for the parametric submodel, focusing on the first of these problems. To the best of our knowledge, this is the first study investigating robustness of TMLE to different specifications of the parametric submodel. We find that the choice of submodel can have an important impact on the behavior of the estimator in finite samples.