• casual inference
• conditional average treatment effect
• doubly-robust
• functional data analysis
• meta-learners

This work addresses the estimation of the Functional Conditional Average Treatment Effect (F-CATE )
– that is, the causal effect of a treatment on a functional outcome, conditional on covariates – in the
framework of Functional Data Analysis (FDA). The objective is to uncover how treatment effects evolve
over time and vary across levels of observed characteristics, a central challenge in modern causal inference
with functional responses.
To tackle this problem, we propose a novel class of functional meta-learners, specifically adapting
R- and DR-learners to the functional setting, and emphasizing doubly robust (DR) approaches that
combine outcome regression and propensity modeling to obtain unbiased estimators, even under partial
mis-specification. Our methodology is grounded in a rigorous theoretical analysis, providing convergence
guarantees and mathematical justification for the robustness properties of the estimators.
We conduct extensive simulation studies to assess performance under various nuisance function mis-
specification scenarios, illustrating the double robustness of the DR-learner: accurate recovery of F-CATE
is achieved as long as either the propensity model or the outcome model is correctly specified.
Finally, we apply our estimators to real-world data from the Survey of Health, Ageing and Retirement
in Europe (SHARE), evaluating the causal effect of chronic medical conditions on the quality-of-life
trajectories of older adults. The analysis controls for a rich set of covariates and provides visualizations
of treatment effects across different subpopulations. Results demonstrate the practical value of functional
meta-learners in capturing treatment heterogeneity over time.