Locally correct confidence intervals for discrete sampling distributions

When a sampling distribution is discrete, the coverage of a confidence interval follows a sequence of peaks and troughs when plotted against the value of the parameter of interest. Then methods of forming a confidence interval must either be conservative, with a coverage that is almost always above the nominal confidence level, or give a coverage that is sometimes below the nominal level. Many methods have been proposed that adopt the latter policy, so the requirements of a confidence interval are often tempered. Garthwaite et al. (2024) suggest that the definition of a confidence interval should be relaxed in a clearly defined way if the definition is not strictly adhered to. For one‐sided intervals, they define locally correct confidence intervals as intervals whose average coverage between consecutive peaks is no smaller than the nominal level. Two‐sided locally correct confidence intervals are formed from two one‐sided intervals. They propose a method for forming locally correct intervals for a binomial proportion that minimizes the average length of such intervals. In this article, we generalize the method so that it can be used with other discrete sampling models. The method is illustrated through examples and compared with alternative methods.