B0205
Title: One-step estimation of differentiable Hilbert-valued parameters
Authors: Alex Luedtke - University of Washington (United States) [presenting]
Incheoul Chung - University of Washington (United States)
Abstract: Estimators are presented for smooth Hilbert-valued parameters, where smoothness is characterized by a pathwise differentiability condition. When the parameter space is a reproducing kernel Hilbert space, a means is provided to obtain efficient, root-n rate estimators and corresponding confidence sets. These estimators are generalizations of cross-fitted one-step estimators based on Hilbert-valued efficient influence functions. Theoretical guarantees are given even when arbitrary estimators of nuisance functions are used, including machine-learning-based ones. These results naturally extend to Hilbert spaces that lack a reproducing kernel, as long as the parameter has an efficient influence function. However, the unfortunate fact is also uncovered that, when there is no reproducing kernel, many interesting pathwise differentiable parameters fail to have an efficient influence function. For these cases, a regularized one-step estimator is proposed with associated confidence sets. Pathwise differentiability, which is a central requirement of the approach, holds in many cases. Specifically, multiple examples of pathwise differentiable parameters are provided and corresponding estimators and confidence sets are developed. Among these examples, four are particularly relevant to ongoing research in causal inference: the counterfactual density function, dose-response function, conditional average treatment effect function, and counterfactual kernel mean embedding.
