Optimal Stopping for Interval Estimation in Bernoulli Trials

We propose an optimal sequential methodology for obtaining confidence intervals for a binomial proportion θ. Assuming that an i.i.d. random sequence of Benoulli(θ) trials is observed sequentially, we are interested in designing a)~a stopping time T that will decide when is the best time to stop sampling the process, and b)~an optimum estimator hatθT that will provide the optimum center of the interval estimate of θ. We follow a semi-Bayesian approach, where we assume that there exists a prior distribution for θ, and our goal is to minimize the average number of samples while we guarantee a minimal coverage probability level. The solution is obtained by applying standard optimal stopping theory and computing the optimum pair (T,hatθT) numerically. Regarding the optimum stopping time component T, we demonstrate that it enjoys certain very uncommon characteristics not encountered in solutions of other classical optimal stopping problems. Finally, we compare our method with the optimum fixed-sample-size procedure but also with existing alternative sequential schemes.