Doubly-Robust Functional Average Treatment Effect Estimation

Understanding causal relationships in the presence of complex, structured data remains a central challenge in modern statistics and science in general. While traditional causal inference methods are well-suited for scalar outcomes, many scientific applications demand tools capable of handling functional data – outcomes observed as functions over continuous domains such as time or space. Motivated by this need, we propose DR-FoS, a novel method for estimating the Functional Average Treatment Effect (FATE) in observational studies with functional outcomes. DR-FoS exhibits double robustness properties, ensuring consistent estimation of FATE even if either the outcome or the treatment assignment model is misspecified. By leveraging recent advances in functional data analysis and causal inference, we establish the asymptotic properties of the estimator, proving its convergence to a Gaussian process. This guarantees valid inference with simultaneous confidence bands across the entire functional domain. Through extensive simulations, we show that DR-FoS achieves robust performance under a wide range of model specifications. Finally, we illustrate the utility of DR-FoS in a real-world application, analyzing functional outcomes to uncover meaningful causal insights in the SHARE ({\em Survey of Health, Aging and Retirement in Europe}) dataset.

💡 Research Summary

The paper introduces DR‑FoS (Doubly‑Robust Functional‑on‑Scalar), a novel estimator for the Functional Average Treatment Effect (FATE) in observational studies where outcomes are functions defined over a continuous domain (e.g., time or space). Traditional causal inference methods focus on scalar outcomes, while many modern scientific applications generate functional data that require methods capable of handling infinite‑dimensional structures. Existing functional approaches either ignore covariates or rely on models that are not robust to misspecification. DR‑FoS fills this gap by extending the double‑robust paradigm to functional outcomes within a Banach‑space framework.

Key methodological components are:

1. Banach‑space formulation – Outcomes (Y_i) are treated as elements of the space (\mathcal{C}(T)) of continuous functions on a closed interval (T), equipped with the sup‑norm (|f|_{\infty}). This choice enables pointwise control and the construction of simultaneous confidence bands, which would be problematic under an (L^2) norm.

2. Double‑robust construction – The propensity score (\pi_a(x)=P(A=a\mid X=x)) and the conditional mean function (\mu_a(x)=E