Tandem-width sequential confidence intervals for a Bernoulli proportion

We propose a two-stage sequential method for obtaining tandem-width confidence intervals for a Bernoulli proportion p. The term “tandem-width” refers to the fact that the half-width of the 100(1 - α)% confidence interval is not fixed beforehand; it is instead required to satisfy two different half-width upper bounds, h₀ and h₁, depending on the (unknown) values of p. To tackle this problem, we first propose a simple but useful sequential method for obtaining fixed-width confidence intervals for p, whose stopping rule is based on the minimax estimator of p. We observe Bernoulli(p) trials sequentially, and for some fixed half-width h = h₀ or h₁, we develop a stopping time T such that the resulting confidence interval for p, [(Formula presented.)], covers the parameter with confidence at least 100(1 - α)% where (Formula presented.) is the maximum likelihood estimator of p at time T. Furthermore, we derive theoretical properties of our proposed fixed-width and tandem-width methods and compare their performances with existing alternative sequential schemes. The proposed minimax-based fixed-width method performs similarly to alternative fixed-width methods, while being easier to implement in practice. In addition, the proposed tandem-width method produces effective savings in sample size compared to the fixed-width counterpart and provides excellent results for scientists to use when no prior knowledge of p is available.