Paper: Treatment Effects Estimation By Uniform Transformer (ICLR-2024)

Reviewer comment (Theory):
The use of a uniform transformer is not well-motivated. It is not clear to me, from the paper, why a uniform transformation is preferred over an estimator with densities \( f_T(X) \) and \( f_C(X) \) estimated separately without any transformation. Is it because it is harder to study such an estimator and thus harder to obtain the optimal bandwidth?

Author response:
This is a great question. Because of the uniform transformer, the estimator for $\mu_{CT}$ is just a simple U-statistics, so we can choose a suitable tuning parameter (which also relies on the smoothness level of the response function) to reduce bias. Whether this idea can extend if we estimate $f_T(X)$ and $f_C(X)$ separately is unclear. If we have an estimator for $f_C(X)$, a potential estimator is
$$
\hat{\mu}_{CT}={\sum_{i_1,i_2=1}^nY_{i_1}(1-Z_{i_1})\hat{f}_C^{-1}(X_{i_1})K_H(X_{i_1}- X_{i_2})Z_{i_2} \over \sum_{i_1,i_2=1}^n (1-Z_{i_1})\hat{f}_C^{-1}(X_{i_1})K_H(X_{i_1}- X_{i_2})Z_{i_2}}
$$
The theoretical analysis can be different. This direction deserves further study in the future.

Another benefit of using a transformer is that Rosenblatt’s transformation relies only on the cumulative distribution function. In some applications, estimating the (conditional) cumulative distribution function is easier than the probability density function. For example, when $d=1$, the uniform transformer can be estimated by an empirical cumulative distribution function, and we do not need to estimate the probability density function.

Reviewer comment (Theory):
The upper bound in Theorem 1 still relies largely on having accurate knowledge of \( f_C \) (and also its smoothness level due to the \( \alpha,\beta<\gamma \) constraint), and it doesn’t seem to me that the proposed method with the adaptive uniform transformer is able to achieve the minimax rate in general.

Author response:
Thanks for raising this good point. The proof of Theorem 2 suggests that whether $f_C(X)$ is known in advance or not, the minimax rate of estimating $\mu_{CT}$ is the same. In addition, the existing minimax optimal estimators also need extra assumptions to achieve the minimax rate. For example, in Robins et al. (2017), Section 3.1 suggests that the higher order influence estimator also needs the function g to be known or easy to estimate (e.g., smooth enough). It is still an open question if the minimax optimal rate can be achieved without these extra assumptions. This is an interesting future direction.

Reviewer comment (Theory):
The minimax optimality results are only for a very restricted class of problems.

Author response:
This is a valid point. In this paper, we only focus on the estimation of the average treatment effect and show the feasibility of a simple weighting estimator, demonstrating the potential of this idea. We will continue working in this direction and see whether we can extend the idea in simple U-statistics to more general problems.

Reviewer comment (Theory):
In general, what is the rate that can be achieved using the adaptive transformer? When adding strong enough regularity conditions, can it in general achieve the rate specified in Corollary 1? And what would those conditions be like?

Author response:
This is a great question. The exact rate can be found at the end of step 1 in Proof of Theorem 3 (Section D.5). If the adaptive transformer is estimated from a data set with sample size in control $n_{0}$, the condition we need is
$$
n_0^{-2\alpha\over d}\le n_1^{-4(\alpha+\beta)\over d+2((\alpha+\beta)}.
$$
It is easier to satisfy when the dimension $d$ is smaller.