  Randomized controlled trials (RCTs) are the gold standard for evaluating the
causal effect of a treatment; however, they often have limited sample sizes and
sometimes poor generalizability. On the other hand, non-randomized,
observational data derived from large administrative databases have massive
sample sizes and better generalizability, but they are prone to unmeasured
confounding bias. It is thus of considerable interest to reconcile effect
estimates obtained from randomized controlled trials and observational studies
investigating the same intervention, potentially harvesting the best from both
realms. In this paper, we theoretically characterize the potential efficiency
gain of integrating observational data into the RCT-based analysis from a
minimax point of view. For estimation, we derive the minimax rate of
convergence for the mean squared error, and propose a fully adaptive anchored
thresholding estimator that attains the optimal rate up to poly-log factors.
For inference, we characterize the minimax rate for the length of confidence
intervals and show that adaptation (to unknown confounding bias) is in general
impossible. A curious phenomenon thus emerges: for estimation, the efficiency
gain from data integration can be achieved without prior knowledge on the
magnitude of the confounding bias; for inference, the same task becomes
information-theoretically impossible in general. We corroborate our theoretical
findings using simulations and a real data example from the RCT DUPLICATE
initiative [Franklin et al., 2021b].
