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I consider estimation of the average treatment effect (ATE), in a populationcomposed of $S$ groups or units, when one has unbiased estimators of eachgroup's conditional average treatment effect (CATE). These conditions are metin stratified experiments and in matching studies. I assume that each CATE isbounded in absolute value by $B$ standard deviations of the outcome, for someknown $B$. This restriction may be appealing: outcomes are often standardizedin applied work, so researchers can use available literature to determine aplausible value for $B$. I derive, across all linear combinations of the CATEs'estimators, the minimax estimator of the ATE. In two stratified experiments, myestimator has twice lower worst-case mean-squared-error than the commonly-usedstrata-fixed effects estimator. In a matching study with limited overlap, myestimator achieves 56\% of the precision gains of a commonly-used trimmingestimator, and has an 11 times smaller worst-case mean-squared-error.

Trading-off Bias and Variance in Stratified Experiments and in Matching  Studies, Under a Boundedness Condition on the Magnitude of the Treatment  Effect