A modified rule of three for the one-sided binomial confidence interval

Abstract

Consider the one-sided binomial confidence interval $\left(L,1\right)$ containing the unknown parameter p when all n trials are successful, and the significance level α to be five or one percent. We develop two functions (one for each level) that represent approximations within $\alpha /\sqrt{3}$ of the exact lower-bound L = α ^1/n . Both the exponential (referred to as a modified rule of three) and the logarithmic function are shown to outperform the standard rule of three L ≃ 1 - 3/n over each of their respective ranges, that together encompass all sample sizes n ≥ 1. Specifically for the exponential, we find that $\exp \left(-3/n\right)$ is a better lower bound when α = 0.05 and n < 1054 and that $\exp \left(-4.6569/n\right)$ is a better bound when α = 0.01 and n < 209.

Keywords: 62F25; 90C30; exponential approximation; logarithmic approximation; one-sided binomial confidence interval; optimization; rule of three.