  Functional data often arise in the areas where the causal treatment effect is
of interest. However, research concerning the effect of a functional variable
on an outcome is typically restricted to exploring the association rather than
the casual relationship. The generalized propensity score, often used to
calibrate the selection bias, is not directly applicable to a functional
treatment variable due to a lack of definition of probability density function
for functional data. Based on the functional linear model for the average
dose-response functional, we propose three estimators, namely, the functional
stabilized weight estimator, the outcome regression estimator and the doubly
robust estimator, each of which has its own merits. We study their theoretical
properties, which are corroborated through extensive numerical experiments. A
real data application on electroencephalography data and disease severity
demonstrates the practical value of our methods.
