  We study treatment effect estimation with functional treatments where the
average potential outcome functional is a function of functions, in contrast to
continuous treatment effect estimation where the target is a function of real
numbers. By considering a flexible scalar-on-function marginal structural
model, a weight-modified kernel ridge regression (WMKRR) is adopted for
estimation. The weights are constructed by directly minimizing the uniform
balancing error resulting from a decomposition of the WMKRR estimator, instead
of being estimated under a particular treatment selection model. Despite the
complex structure of the uniform balancing error derived under WMKRR,
finite-dimensional convex algorithms can be applied to efficiently solve for
the proposed weights thanks to a representer theorem. The optimal convergence
rate is shown to be attainable by the proposed WMKRR estimator without any
smoothness assumption on the true weight function. Corresponding empirical
performance is demonstrated by a simulation study and a real data application.
