Confidence intervals with control of the sign error in low power settings

When hypothesis tests of H0 : θ = 0 have low power, it is possible that their rejection can frequently be accompanied by an estimate θ̂ that has the wrong sign and significantly exaggerated magnitude. Such sign errors are less likely when the confidence interval for θ is well separated from 0, as measured in units of the confidence interval’s width. Sign errors can be controlled by declaring confidence in the sign of θ only when H0 is also rejected at a smaller level α2 = 2α1αS , where αS ≤ 1/2 is a user specified upper bound on the probability of a sign error given that H0 has been rejected. This procedure is a very simple form of inference after model selection: the selected model is that θ 6= 0 and the second inference is then on sign(θ).