Improving the false coverage rate adjusted confidence intervals

Abstract This study addresses the challenges of inference following selection in fields like clinical trials, genome-wide association studies, and functional magnetic resonance imaging, where traditional methods like simultaneous confidence intervals (CIs) might be too conservative. We introduce an improved false coverage-statement rate controlling CIs, when the selection is done by passing a threshold in a certain direction. The CIs for the selected parameters are similar to those proposed by Benjamini and Yekutieli (2005) on the inward end, and to the standard nonadjusted CIs on the outward end. The centre of the suggested CI is a shrunk estimator of the selected parameter. This simple improvement is uniformly better for the one-directional selection, and we also suggest how to apply it for a two-directional selection. We prove that the false coverage-statement control for independent estimators and provide simulation evidence for its robustness under dependency.