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