A generalized Agresti–Coull type confidence interval for a binomial proportion

One of the fundamental topics in statistical inference is constructing a confidence interval for a binomial proportion p. It is well known that commonly used asymptotic confidence intervals, such as the Wilson and Agresti–Coull confidence intervals, suffer from systematic bias and oscillations in their coverage probabilities. We generalize asymptotic confidence intervals, including the Wald, Wilson and Agresti–Coull intervals, and propose a generalized Agresti–Coull type confidence interval by adjusting the bias with the saddlepoint approximation. We compare the coverage probabilities and lengths of the proposed confidence interval with those of other popular asymptotic confidence intervals. We show that the proposed confidence interval is more stable than the Wilson interval at the boundaries of p and has a shorter length than the Agresti–Coull interval.