Adaptive confidence bands for shape-constrained causal continuous treatment effect curves

Thomas M. Menino Convention & Exhibition Center 

In causal inference with observational data when treatment is continuous rather than discrete, new inferential targets arise. Here we form (simultaneous) confidence bands over the continuous treatment effect curve. We form confidence bands based on assuming the curve satisfies shape constraints such as monotonicity, under the assumption of no unmeasured confounding. The assumption of monotonicity (or unimodality) is often a natural one for the treatment effect curve, at least over a reasonable domain of treatment. As is commonly the case, two nuisance functions arise, and our bands are ``doubly robust'' in the sense that they are asymptotically valid if the product of the error rates of the two nuisance functions is small enough. Furthermore, our bands are adaptive to a range of function smoothness values in that when the true smoothness level is unknown the bands still achieve the same (optimal) rate of convergence (in sup-norm) as when the true smoothness level is known. If time permits we will also develop analogous confidence bands under the constraint of concavity.