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. Comparison of such methods is tricky -- the best method should perhaps be the one that gives the shortest intervals (on average), but when is the coverage of a method so poor that it should not be classed as a means of forming 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; methods which meet the criterion are said to yield $\textit{locally correct confidence intervals}$. 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. The mid-$p$ method also satisfies the new criterion and has its own optimality property.