We propose an optimal sequential methodology for obtaining confidence intervals for a binomial proportion θ. Assuming that an independent and identically distributed sequence of Bernoulli (θ) trials is observed sequentially, we are interested in designing: 1) a stopping time T that will decide the best time to stop sampling the process and 2) an optimum estimator θ̂T 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 specified coverage probability level. The solution is obtained by applying standard optimal stopping theory and computing the optimum pair (T, θ̂T) numerically. Regarding the optimum stopping time component T, we demonstrate that it enjoys certain very interesting characteristics not commonly encountered in solutions of other classical optimal stopping problems. In particular, we prove that, for a particular prior (beta density), the optimum stopping time is always bounded from above and below; it needs to first accumulate a sufficient amount of information before deciding whether or not to stop, and it will always terminate before some finite deterministic time. We also conjecture that these properties are present with any prior. Finally, we compare our method with the optimum fixed-sample-size procedure as well as with existing alternative sequential schemes.

We consider the problem of parameter estimation under a sequential framework. Specifically we assume that an i.i.d. random process is observed sequentially with its common pdf having a random parameter that must be estimated. We are interested in designing a stopping time that will decide when is the best moment to stop sampling the process and an estimator that will use the acquired samples in order to provide the desired estimate. We follow a semi-Bayesian approach where we assign cost to the pair (estimate, true parameter) and our goal is to minimize the average sample size guaranteeing at the same time an average cost below some prescribed level. For our analysis we adopt a conditional average cost which leads to a considerable simplification in the sequential estimation problem, otherwise known to be analytically intractable. We apply our results to a number of examples and compare our method with the optimum fixed sample size but also with existing sequential schemes.