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 but typically rely on fixed error margins that do not scale with the magnitude of the proportion, thus distorting uncertainty quantification at the extremes. As an alternative method to reduce these boundary distortions, we propose a novel hybrid approach that 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 regularisation term that is inversely proportional to sample size. Extensive simulation studies and real-data applications demonstrate that the proposed method consistently achieves competitive or superior coverage proportions with narrower or more conservative interval widths compared to Jeffreys and Wilson score intervals, especially for rare and extreme events. Geometric analysis of the tuning curves reveals convex-to-linear transitions and mirrored symmetry across the rare-extreme spectrum, which underscores its boundary sensitivity and adaptivity. Our method offers a theoretically grounded, computationally efficient and practically robust estimation of rare-event intervals which has applications in safety-critical reliability, epidemiology, and early-phase clinical trials.