Simultaneous Inference for Proximal Causal Learning with Weak Proxies
Proximal causal learning uses the treatment-inducing proxy and the outcome-inducing proxy through the outcome bridge and the treatment bridge to provide a way to address unobserved confounding in observational studies. This paper studies simultaneous inference for the marginal counterfactual distribution and quantile treatment effect under weak proxies. First, we propose a measure of weak-proxy strength and tightens the population-bias bound for the counterfactual CDF estimate. The baseline Cauchy--Schwarz bound introduces the inverse of weak-proxy strength twice, whereas the proposed bound introduces the inverse of weak-proxy strength only once and is sharp for the bridge estimators, thereby relaxing the restriction on the decay rate of weak-proxy strength. Second, we prove that, even when outcome thresholds vary continuously, the number of distinct estimated outcome bridges is at most one greater than the number of observations in the training sample, thereby establishing a uniform expansion across treatment arms and outcome thresholds. Finally, we prove that the critical value remains valid after bridge estimation, thereby obtaining simultaneous CDF bands, and constructs quantile treatment effect bands through confidence-band inversion. The proposed method provides deterministic band-width bounds and does not require counterfactual density estimation. In the numerical study, the baseline Cauchy--Schwarz bound overestimates the true population bias by a factor of approximately 50, whereas the proposed bound is very close to the true population bias. The quantile treatment effect bands constructed by the proposed method achieve coverage of 99.8\%--100.0\% while being only approximately 20\% wider than the oracle 95\% bands, compared with baseline coverage of 68.4\%--89.2\%.