Estimation of a probability with optimum guaranteed confidence in inverse binomial sampling

Sequential estimation of a probability p by means of inverse binomial sampling is considered. For μ1, μ2 > 1 given, the accuracy of an estimator p̂ is measured by the confidence level Pr[p/μ2 ≤ p̂ ≤ pμ1]. The confidence levels c0 that can be guaranteed for p unknown, i.e. such that Pr[p/μ2 ≤ p̂ ≤ pμ1] ≥ c0 for all p ∈ (0, 1), are investigated. It is shown that, within the general class of randomized or nonrandomized estimators based on inverse binomial sampling, there is a maximum c0 that can be guaranteed for p arbitrary. A nonrandomized estimator is given that achieves this maximum guaranteed confidence under mild conditions on μ1, μ2.