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

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-p 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.