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Several problems in statistics involve the combination of high-varianceunbiased estimators with low-variance estimators that are only unbiased understrong assumptions. A notable example is the estimation of causal effects whilecombining small experimental datasets with larger observational datasets. Thereexist a series of recent proposals on how to perform such a combination, evenwhen the bias of the low-variance estimator is unknown.  To build intuition for the differing trade-offs of competing approaches, weargue for examining the finite-sample estimation error of each approach as afunction of the unknown bias. This includes understanding the bias threshold --the largest bias for which a given approach improves over using the unbiasedestimator alone. Though this lens, we review several recent proposals, andobserve in simulation that different approaches exhibits qualitativelydifferent behavior.  We also introduce a simple alternative approach, which compares favorably insimulation to recent alternatives, having a higher bias threshold and generallymaking a more conservative trade-off between best-case performance (when thebias is zero) and worst-case performance (when the bias is adversariallychosen). More broadly, we prove that for any amount of (unknown) bias, the MSEof this estimator can be bounded in a transparent way that depends on thevariance / covariance of the underlying estimators that are being combined.

Understanding the Risks and Rewards of Combining Unbiased and Possibly Biased Estimators, with Applications to Causal Inference