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 $μ_1,μ_2>1$ given, the accuracy of an estimator $\hat{p}$ is measured by the confidence level $P[p/μ_2\leq\hat{p}\leq pμ_1]$. The confidence levels $c_0$ that can be guaranteed for $p$ unknown, that is, such that $P[p/μ_2\leq \hat{p}\leq pμ_1]\geq c_0$ for all $p\in(0,1)$, 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 $c_0$ that can be guaranteed for arbitrary $p$. A non-randomized estimator is given that achieves this maximum guaranteed confidence under mild conditions on $μ_1$, $μ_2$.