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

Abstract

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