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Score-Preserving Targeted Maximum Likelihood Estimation

Noel Pimentel, Alejandro Schuler, Mark van der Laan

TL;DR

Score-Preserving TMLE (SP-TMLE) extends targeted maximum likelihood estimation by identifying and preserving the scores solved by an initial estimator and updating along a multi-dimensional submodel that spans both the efficient influence function and these scores. This approach aims to reduce the remainder term in the von Mises expansion, improving finite-sample bias and variance, and it is demonstrated via simulations using HAL as the initial fit, showing reduced bias relative to plug-in HAL and reduced variance with respect to vanilla TMLE, along with better standard error estimation and coverage. The method couples TMLE with explicit score equations, offering a practical best-of-both-worlds performance in small samples, while acknowledging reduced generality due to the need for explicit score structures. Overall, SP-TMLE blends the robustness of TMLE with enhanced efficiency from score-solving, providing a promising tool for finite-sample semiparametric estimation, particularly in treatment-specific mean settings.

Abstract

Targeted maximum likelihood estimators (TMLEs) are asymptotically optimal among regular, asymptotically linear estimators. In small samples, however, we may be far from "asymptopia" and not reap the benefits of optimality. Here we propose a variant (score-preserving TMLE; SP-TMLE) that leverages an initial estimator defined as the solution of a large number of possibly data-dependent score equations. Instead of targeting only the efficient influence function in the TMLE update to knock out the plug-in bias, we also target the already-solved scores. Solving additional scores reduces the remainder term in the von-Mises expansion of our estimator because these scores may come close to spanning higher-order influence functions. The result is an estimator with better finite-sample performance. We demonstrate our approach in simulation studies leveraging the (relaxed) highly adaptive lasso (HAL) as our initial estimator. These simulations show that in small samples SP-TMLE has reduced bias relative to plug-in HAL and reduced variance relative to vanilla TMLE, blending the advantages of the two approaches. We also observe improved estimation of standard errors in small samples.

Score-Preserving Targeted Maximum Likelihood Estimation

TL;DR

Score-Preserving TMLE (SP-TMLE) extends targeted maximum likelihood estimation by identifying and preserving the scores solved by an initial estimator and updating along a multi-dimensional submodel that spans both the efficient influence function and these scores. This approach aims to reduce the remainder term in the von Mises expansion, improving finite-sample bias and variance, and it is demonstrated via simulations using HAL as the initial fit, showing reduced bias relative to plug-in HAL and reduced variance with respect to vanilla TMLE, along with better standard error estimation and coverage. The method couples TMLE with explicit score equations, offering a practical best-of-both-worlds performance in small samples, while acknowledging reduced generality due to the need for explicit score structures. Overall, SP-TMLE blends the robustness of TMLE with enhanced efficiency from score-solving, providing a promising tool for finite-sample semiparametric estimation, particularly in treatment-specific mean settings.

Abstract

Targeted maximum likelihood estimators (TMLEs) are asymptotically optimal among regular, asymptotically linear estimators. In small samples, however, we may be far from "asymptopia" and not reap the benefits of optimality. Here we propose a variant (score-preserving TMLE; SP-TMLE) that leverages an initial estimator defined as the solution of a large number of possibly data-dependent score equations. Instead of targeting only the efficient influence function in the TMLE update to knock out the plug-in bias, we also target the already-solved scores. Solving additional scores reduces the remainder term in the von-Mises expansion of our estimator because these scores may come close to spanning higher-order influence functions. The result is an estimator with better finite-sample performance. We demonstrate our approach in simulation studies leveraging the (relaxed) highly adaptive lasso (HAL) as our initial estimator. These simulations show that in small samples SP-TMLE has reduced bias relative to plug-in HAL and reduced variance relative to vanilla TMLE, blending the advantages of the two approaches. We also observe improved estimation of standard errors in small samples.

Paper Structure

This paper contains 13 sections, 13 equations, 1 figure.

Table of Contents

  1. Notation and Preliminaries
  2. Setup
  3. Efficiency Theory
  4. Targeted Maximum Likelihood Estimation
  5. Score-Solving Initial Estimators
  6. Score-Preserving TMLE
  7. SP-TMLE for the Treatment-Specific Mean
  8. Simulation Study
  9. Data-Generating Process
  10. Estimators
  11. Results
  12. Discussion
  13. Submodel Scores

Figures (1)

  • Figure 1: Bias, variance, and MSE of SP-TMLE, TMLE, relaxed HAL and HAL. Bias was calculated by taking the mean over each 500 estimates of each estimator and subtracting the true parameter value $\Psi(P_0) = 0.5$. Variance was calculated by taking the variance over each 500 estimates of each estimator. 95% confidence intervals were estimated by taking the empirical variance of the estimated influence function (and dividing by $n$) for each of the 500 estimates, and was used to assess coverage.