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

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

2008-09-15

arxiv.org/abs/0809.2402v4

Bernoulli 2010, Vol. 16, No. 2, 493-513

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.3150/09-BEJ219 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/09-BEJ219

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 $\mu_1$, $\mu_2$.

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