Locally correct confidence intervals for a binomial proportion: A new criteria for an interval estimator

Abstract Well‐recommended methods of forming “confidence intervals” for a binomial proportion give interval estimates that do not actually meet the definition of a confidence interval, in that their coverages are sometimes lower than the nominal confidence level. The methods are favoured because their intervals have a shorter average length than the Clopper–Pearson (gold‐standard) method, whose intervals really are confidence intervals. As the definition of a confidence interval is not being adhered to, another criterion for forming interval estimates for a binomial proportion is needed. In this paper, we suggest a new criterion for forming one‐sided intervals and equal‐tail two‐sided intervals. Methods which meet the criterion are said to yield locally correct confidence intervals . We propose a method that yields such intervals and prove that its intervals have a shorter average length than those of any other method that meets the criterion. Compared with the Clopper–Pearson method, the proposed method gives intervals with an appreciably smaller average length. For confidence levels of practical interest, the mid‐ method also satisfies the new criterion and has its own optimality property. It gives locally correct confidence intervals that are only slightly wider than those of the new method.