  Sequential estimation of a probability $p$ by means of inverse binomial
sampling is considered. For $\mu_1,\mu_2>1$ given, the accuracy of an estimator
$\hat{p}$ is measured by the confidence level $P[p/\mu_2\leq\hat{p}\leq
p\mu_1]$. The confidence levels $c_0$ that can be guaranteed for $p$ unknown,
that is, such that $P[p/\mu_2\leq \hat{p}\leq p\mu_1]\geq c_0$ 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 $\mu_1$, $\mu_2$.
