Poisson and (Negative) Binomial Confidence Bounds With Applications to Rare Event Probabilities

We present here by direct argument the classical Clopper-Pearson (1934) “exact” confidence bounds for the binomial or negative binomial parameter p, for the Poisson parameter λ, and for the ratio of two Poisson parameters, ρ = λ1/λ2. The bounds presented here are exact in the sense that their confidence level of covering the unknown parameters is at least the specified and targeted value γ, 0 < γ < 1. Because of the discrete nature of the underlying distributions the actual confidence varies with the unknown parameter and can, for some parameter ranges, be considerably higher than the stated value γ. In that sense these bounds are conservative in their coverage. Agresti and Coull (1998) have recently discussed advantages of alternate methods, where the actual coverage oscillates more or less around the target value γ, and not above it. The advantage of such intervals is that they are somewhat shorter than the Clopper-Pearson intervals. Given that we often deal with confidence bounds concerning rare events we take the conservative approach of Clopper and Pearson. A recent discussion for the Poisson parameter can found in Barker (2002). It is shown how such “exact” bounds can be computed quite easily using either the Excel spread sheet or the statistical packages R or S-Plus. We first give the argument for confidence bounds for a Poisson parameter λ. The arguments for lower and upper bounds are completely parallel and it suffices to get a complete grasp of only one such derivation. This is followed by the corresponding argument for the binomial or negative binomial parameter p. For very small p it is pointed out how to use the very effective Poisson approximation to get bounds on p. Finally we give the classical method for constructing confidence bounds on the ratio ρ = λ1/λ2 based on two independent Poisson counts X and Y from Poisson distributions with parameters λ1 and λ2, respectively. This MC: 7L-22, Phone: (425) 865-3623, e-mail: [email protected], http://www.rt.cs.boeing.com/MEA/stat/scholz/ for the latest version of this document