Adaptive Bayesian Interval Estimation for Rare Binomial Events: A Variance-Blending Calibration Framework

Classical binomial interval methods often exhibit poor performance when applied to extreme conditions, such as rare-event scenarios or small-sample estimations. Recent frequentist and Bayesian approaches have improved coverage in small samples and rare events. However, they typically rely on fixed error margins that do not scale with the magnitude of the proportion. This distorts uncertainty quantification at the extremes. As an alternative method to reduce these boundary distortions, we propose a novel hybrid approach. It blends Bayesian, frequentist, and approximation-based techniques to estimate robust and adaptive intervals. The variance incorporates sampling variability, Wilson score margin of error, a tuned credible level, and a gamma regularization term that is inversely proportional to sample size. Extensive simulation studies and real-data applications demonstrate that the proposed method consistently achieves better coverage proportions at all sample sizes and proportions. It provides more conservative interval widths below a sample size of 50 and competitively narrower widths from moderate to large sample sizes, especially beyond 50, compared to the Jeffreys’ and Wilson score intervals. Geometric analysis of the tuning curves demonstrates how the blended method adaptively tunes credible levels across binomial extremes. It starts at higher values for small samples and gradually flattens into near-linear, symmetric trajectories as sample size increases. This ensures robust coverage and balanced sensitivity. Our method offers a theoretically grounded, computationally efficient, and practically robust estimation of rare-event intervals. These intervals have applications in safety-critical reliability, epidemiology, and early-phase clinical trials.