An optimal exact confidence interval for the difference of two independent binomial proportions

The difference between two proportions is the most important parameter in comparing two treatments based on independent two binomials and has garnered widespread application across various fields, particularly in clinical trials. There exists significant interest in devising optimal confidence intervals for the difference. Approximate intervals relying on asymptotic normality may lack reliability, thus calling for enhancements in exact confidence interval construction to bolster reliability and precision. In this paper, we present a novel approach that leverages the most probable test statistic and employs the $$-function method to construct an optimal exact interval for the difference. We juxtapose the proposed interval against other exact intervals established through methodologies such as the Agresti-Min exact unconditional method, the Wang method, the fiducial method, and the hybrid score method. Our comparative analysis, employing the infimum coverage probability and total interval length as evaluation metrics, underscores the uniformly superior performance of the proposed interval. Additionally, we elucidate the application of these exact intervals using two real datasets.