Functional structural equation models with out-of-sample guarantees

Statistical learning methods typically assume that the training and test data originate from the same distribution, enabling effective risk minimization. However, real-world applications frequently involve distributional shifts, leading to poor model generalization. To address this, recent advances in causal inference and robust learning have introduced strategies such as invariant causal prediction and anchor regression. While these approaches have been explored for traditional structural equation models (SEMs), their extension to functional systems remains limited. This paper develops a risk minimization framework for functional SEMs using linear, potentially unbounded operators. We introduce a functional worst-risk minimization approach, ensuring robust predictive performance across shifted environments. Our key contribution is a novel worst-risk decomposition theorem, which expresses the maximum out-of-sample risk in terms of observed environments. We establish conditions for the existence and uniqueness of the worst-risk minimizer and provide consistent estimation procedures. Empirical results on functional systems illustrate the advantages of our method in mitigating distributional shifts. These findings contribute to the growing literature on robust functional regression and causal learning, offering practical guarantees for out-of-sample generalization in dynamic environments.