Confidence Bounds & Intervals for Parameters Relating to the Binomial, Negative Binomial, Poisson and Hypergeometric Distributions With Applications to Rare Events

We present here by direct argument the classical Clopper-Pearson (1934) “exact” confidence bounds and corresponding intervals for the parameter p of the binomial distribution. The same arguments can be applied to derive confidence bounds and intervals for the negative binomial parameter p, for the Poisson parameter λ, for the ratio of two Poisson parameters, ρ = λ1/λ2, and for the parameter D of the hypergeometric distribution. The 1-sided bounds presented here are exact in the sense that their minimum probability of covering the respective unknown parameter is equal to the specified target confidence level γ = 1−α, 0 < γ < 1 or 0 < α < 1. If θ̂L and θ̂U denote the respective lower and upper confidence bounds for the parameter θ of interest, this amounts to the following coverage properties