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

TL;DR

This paper investigates the maximum guaranteed confidence levels for probability estimation using inverse binomial sampling, introducing an optimal non-randomized estimator that achieves this maximum under certain conditions.

Contribution

It identifies the highest guaranteed confidence level achievable with inverse binomial sampling and proposes an estimator that attains this maximum.

Findings

Maximum guaranteed confidence level is established.

A non-randomized estimator achieving this maximum is proposed.

The results hold under mild conditions on accuracy parameters.

Abstract

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 $\mu_1$, $\mu_2$.