Inference on winners

Many empirical questions concern target parameters selected through optimization. For example, researchers may be interested in the effectiveness of the best policy found in a randomized trial, or the best-performing investment strategy based on historical data. Such settings give rise to a winner's curse, where conventional estimates are biased and conventional confidence intervals are unreliable. This paper develops optimal confidence intervals and median-unbiased estimators that are valid conditional on the target selected and so overcome this winner's curse. If one requires validity only on average over targets that might have been selected, we develop hybrid procedures that combine conditional and projection confidence intervals to offer further performance gains relative to existing alternatives.