New application of the Clopper–Pearson method: a unified framework for extreme event proportion estimation

The estimation of extreme event proportions, such as zero-failure or all-success outcomes, has become increasingly important with advances in medical technology. In such settings, the classical maximum likelihood estimator (MLE) often yields boundary values and clinically impractical results, motivating numerous alternative approaches. However, these methods are largely ad hoc and lack a unified framework. This study introduces a unified methodological framework in which several commonly used extreme-event estimators are reinterpreted as one-sided Clopper–Pearson confidence limits at specific α levels, establishing a transparent mapping between existing estimators and the Clopper–Pearson method. Within this framework, previously proposed estimators are shown to correspond exactly or approximately to one-sided Clopper–Pearson limits in zero-failure or all-success cases, with relationships derived using Taylor expansion and limit arguments. The practical implications of Clopper–Pearson–based point estimation are discussed, including the impact of α on sensitivity, specificity, predictive values, and likelihood ratios. A formula is provided for sample size determination to control deviations from the MLE. Analyses of real diagnostic test data demonstrate that Clopper–Pearson based estimates yield more interpretable and clinically meaningful results than the MLE under extreme-event scenarios.