[Paper Review] Higher Order Targeted Maximum Likelihood Estimation

Mark van der Laan, Zeyi Wang+1 more|arXiv (Cornell University)|Jan 15, 2021

Statistical Methods and Inference26 references4 citations

TL;DR

This paper introduces a higher order targeted maximum likelihood estimator (k-th order TMLE) that improves finite-sample inference by replacing the first-order remainder in standard TMLE with a k+1-th order remainder, leveraging Highly Adaptive Lasso (HAL) regularization to control bias. The method ensures asymptotic linearity and valid inference under weaker regularity conditions, with simulations confirming improved coverage and bias reduction over first-order TMLE.

ABSTRACT

Asymptotic efficiency of targeted maximum likelihood estimators (TMLE) of target features of the data distribution relies on a a second order remainder being asymptotically negligible. In previous work we proposed a nonparametric MLE termed Highly Adaptive Lasso (HAL) which parametrizes the relevant functional of the data distribution in terms of a multivariate real valued cadlag function that is assumed to have finite variation norm. We showed that the HAL-MLE converges in Kullback-Leibler dissimilarity at a rate n-1/3 up till logn factors. Therefore, by using HAL as initial density estimator in the TMLE, the resulting HAL-TMLE is an asymptotically efficient estimator only assuming that the relevant nuisance functions of the data density are cadlag and have finite variation norm. However, in finite samples, the second order remainder can dominate the sampling distribution so that inference based on asymptotic normality would be anti-conservative. In this article we propose a new higher 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+1th order remainder. As a consequence, this k-th order TMLE allows statistical inference only relying on the k+1th order remainder being negligible. We also provide a rationale for the higher order TMLE that it will be superior to the first order TMLE by (iteratively) locally minimizing the exact finite sample remainder of the first order TMLE. The second order TMLE is demonstrated for nonparametric estimation of the integrated squared density and for the treatment specific mean outcome. We also provide an initial simulation study for the second order TMLE of the treatment specific mean confirming the theoretical analysis.

Motivation & Objective

• To address the finite-sample bias in targeted maximum likelihood estimation (TMLE) caused by non-negligible second-order remainders.
• To develop a k-th order TMLE framework that ensures exact linear expansion with a k+1-th order remainder, enabling valid inference without undersmoothing.
• To demonstrate that HAL-regularized higher order TMLE achieves negligible regularization bias, even without undersmoothing, by controlling the L1-norm of the HAL-MLE.
• To provide a finite-sample rationale showing that higher order TMLE minimizes the exact remainder of the first-order TMLE iteratively around the initial estimator.
• To enable inference in settings where the first-order canonical gradient vanishes at the truth, using second-order influence curves.

Proposed method

• Proposes a k-th order TMLE that sequentially targets higher-order fluctuations of the target parameter using HAL-regularized MLEs along least favorable paths.
• Employs a universal least favorable path construction to ensure the score at each step equals the canonical gradient of the fluctuated parameter.
• Derives an exact expansion of the k-th order TMLE in terms of efficient influence functions of higher-order fluctuations, with a remainder of order k+1.
• Controls HAL-regularization bias by bounding the L1-norm of the HAL-MLE, ensuring it is asymptotically negligible without undersmoothing.
• Uses empirical process theory and entropy bounds to establish that the k+1-th order remainder is negligible under minimal smoothness assumptions.
• Provides a constructive algorithm for computing higher-order canonical gradients via adjoint operators and symmetric matrix inversion, enabling automated implementation.

Experimental results

Research questions

• RQ1 Can a k-th order TMLE be constructed such that its remainder is of order k+1, enabling valid inference without relying on undersmoothing?
• RQ2 Does HAL regularization of the nuisance estimators ensure that the regularization bias is negligible in finite samples?
• RQ3 Can higher order TMLE improve coverage and reduce bias compared to first-order TMLE in finite samples?
• RQ4 Is it possible to perform inference when the first-order canonical gradient vanishes at the truth, using second-order influence curves?
• RQ5 Can the higher-order canonical gradients be computed systematically using standard regression or matrix inversion tools?

Key findings

• The k-th order TMLE satisfies an exact linear expansion with a remainder of order k+1, enabling inference based solely on the negligibility of this remainder.
• The HAL-regularization bias is guaranteed to be of small enough order without undersmoothing, due to control of the L1-norm in the HAL-MLE.
• Simulations for nonparametric estimation of integrated squared density and treatment-specific mean outcome confirm that second-order TMLE reduces bias and improves confidence interval coverage compared to first-order TMLE.
• The empirical higher order TMLE achieves performance comparable to undersmoothed versions, making it a robust and recommended choice.
• The method allows inference even when the first-order canonical gradient is zero at the truth, by using the second-order influence curve to obtain a non-degenerate limit distribution.
• A constructive algorithm for computing higher-order canonical gradients is provided using adjoint operators and matrix inversion, enabling practical implementation with standard software.