# Conditions for asymptotic efficiency of TMLE

This post is part of our Q&A series.

A question from graduate students in our Spring 2021 offering of the new course “Targeted Learning in Practice” at UC Berkeley:

## Question:

Hi Mark,

I have a question regarding the requirements for asymptotic efficiency of TMLE.

Asymptotic efficiency of TMLE relies on the second-order remainder being negligible. Is this purely a finite-sample concern, or are there potentially parameters of interest where this isn’t true by construction?

Best,

M.M.

Hi M.M.,

Thank you for the excellent question.

In the past, this was an asymptotic concern as well, since there were no general estimators that achieved the critical $n^{-1/4}$ rate of convergence. Nowadays, we have the highly adaptive lasso (HAL), which achieves $n^{-1/3}$ only assuming that the true function is cadlag with finite sectional variation norm. This type of smoothness assumption is weak, so that this should generally hold making the HAL-TMLE an asymptotically efficient estimator, assuming strong positivity for the target estimand. On the other hand, we could think about problems in which the true function has circular-type discontinuities making it non-cadlag and have infinite variation norm accordingly, or is a continuous function with infinite variation norm. In such cases, I am not aware of any estimator that will get us the desired rate of convergence. Nonetheless, we can still make it work if either $Q(A,W)$ or $g(A \mid W)$ has this desired smoothness condition satisfied. So $Q(A,W)$ might not, but a TMLE that estimates $g(A \mid W)$ with undersmoothed HAL would still be asymptotically efficient if $g(A \mid W)$ is cadlag with finite variation norm. There is also the option to apply HAL to coordinate transformations which could be learned from the data, which would make it possible to approximate the true function with a cadlag function of these transformed covariates. For example, meta-HAL operates like that and would allow to overcome this if given the right type of transformations or learners of transformations.

For sure the exact remainder is a finite sample concern. It has motivated the 1) HAL-based bootstrap for HAL and HAL-TMLE; 2) estimation of nuisance functions with HAL so that it solves lots of score equation, knocking out a lot about the exact remainder; and 3) the higher-order TMLE, reducing the exact remainder into a higher-order difference, to name a few.

Best Wishes,

Mark

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