Confidence Intervals for Selected Parameters

Practical or scientific considerations often lead to selecting a subset of parameters as ``important.'' Inferences about those parameters often are based on the same data used to select them in the first place. That can make the reported uncertainties deceptively optimistic: confidence intervals that ignore selection generally have less than their nominal coverage probability. Controlling the probability that one or more intervals for selected parameters do not cover---the ``simultaneous over the selected'' (SoS) error rate---is crucial in many scientific problems. Intervals that control the SoS error rate can be constructed in ways that take advantage of knowledge of the selection rule. We construct SoS-controlling confidence intervals for parameters deemed the most ``important'' k of m shift parameters because they are estimated (by independent estimators) to be the largest. The new intervals improve substantially over vSid'ak intervals when k is small compared to m, and approach the standard Bonferroni-corrected intervals when k ≈ m. Standard, unadjusted confidence intervals for location parameters have the correct coverage probability for k=1, m=2 if, when the true parameters are zero, the estimators are exchangeable and symmetric.