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One-Step Estimation of Differentiable Hilbert-Valued Parameters
Alex Luedtke1, Incheoul Chung1
1Department of Statistics, University of Washington.
Annals of Statistics
|June 29, 2026
Summary
We developed new statistical estimators for smooth Hilbert-valued parameters, offering efficient estimation and confidence sets even with machine learning nuisance estimators. These methods apply to reproducing kernel Hilbert spaces and beyond, addressing challenges in causal inference.
Area of Science:
- Statistics
- Machine Learning
- Causal Inference
Background:
- Statistical estimation of smooth Hilbert-valued parameters is crucial in various fields.
- Pathwise differentiability is a key condition for developing robust estimators.
- Existing methods may struggle with nuisance parameters or spaces lacking reproducing kernels.
Purpose of the Study:
- To develop efficient, root-n rate estimators and confidence sets for smooth Hilbert-valued parameters.
- To provide theoretical guarantees for these estimators, even when using machine learning techniques for nuisance functions.
- To extend estimation methods to Hilbert spaces without reproducing kernels and address parameters lacking efficient influence functions.
Main Methods:
- Utilizing pathwise differentiability to characterize parameter smoothness.
- Generalizing cross-fitted one-step estimators based on Hilbert-valued efficient influence functions.
- Proposing a regularized one-step estimator for cases without efficient influence functions.
Main Results:
- Achieved efficient, root-n rate estimators and confidence sets in reproducing kernel Hilbert spaces.
- Demonstrated theoretical guarantees for estimators using arbitrary nuisance function estimators, including machine learning methods.
- Extended results to Hilbert spaces lacking reproducing kernels and introduced a novel regularized estimator for challenging cases.
- Confirmed pathwise differentiability for several parameters relevant to causal inference.
Conclusions:
- The proposed estimators offer efficient statistical inference for smooth Hilbert-valued parameters.
- The methods are robust to the use of machine learning for nuisance functions.
- New tools are provided for causal inference, including estimators for counterfactual density and dose-response functions.
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