Optimal Stopping for Interval Estimation in Bernoulli Trials

TL;DR

This paper introduces a semi-Bayesian optimal stopping method for efficiently estimating a binomial proportion with guaranteed coverage, minimizing sampling while ensuring confidence interval accuracy.

Contribution

It develops a novel sequential approach combining optimal stopping theory with Bayesian priors to improve interval estimation efficiency for Bernoulli trials.

Findings

The optimal stopping time exhibits unique properties not seen in classical problems.

The method reduces the average sample size compared to fixed-sample procedures.

It outperforms existing sequential schemes in coverage and efficiency.

Abstract

We propose an optimal sequential methodology for obtaining confidence intervals for a binomial proportion $\theta$. Assuming that an i.i.d. random sequence of Benoulli($\theta$) 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 $\hat{\theta}_{T}$ that will provide the optimum center of the interval estimate of $\theta$. We follow a semi-Bayesian approach, where we assume that there exists a prior distribution for $\theta$, 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 $(T,\hat{\theta}_{T})$ 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.