Beyond exact and asymptotic: optimization-based tuning of confidence intervals for proportions

Abstract

Confidence interval estimation for binomial proportions remains challenging in finite samples due to the discreteness of the data, which induces oscillatory coverage behavior. Classical intervals are often conservative (e.g., Clopper–Pearson) or may exhibit undercoverage (e.g., Wald), and exact nominal coverage cannot be achieved uniformly over the parameter space. We expand Reiczigel’s numerical level adjustment method that treats the quantile or tail probability underlying a confidence interval as a tunable parameter into a general optimization-based framework. By directly targeting the exact coverage function, the tuning parameter is chosen through minimization of a user-specified risk functional, such as mean squared deviation from nominal coverage or absolute deviation of average coverage. Comprehensive numerical investigations across multiple sample sizes and confidence levels show that the tuned intervals reduce global deviations from the nominal level and alleviate conservatism, while making the associated trade-offs in minimum coverage explicit. The comparison with the alternative optimization strategies, including length–coverage optimal (LCO) intervals, highlights the complementary nature of different calibration objectives. Overall, the framework provides a flexible and computationally accessible way to improve the finite-sample calibration of familiar binomial confidence intervals without changing their analytical form.