Tandem-width sequential confidence intervals for a Bernoulli proportion

Abstract

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 $100(1-\alpha )\%$ 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, [${\widehat{p}}_T-h,{\widehat{p}}_T+h$], covers the parameter with confidence at least $100(1-\alpha )\%,$ where ${\widehat{p}}_T$ 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.

KEYWORDS:

• Bernoulli proportion
• binomial distribution
• confidence interval
• sequential analysis
• stopping time

SUBJECT CLASSIFICATIONS:

• 62L12
• 60G40

Acknowledgment

The authors thank the Editor for his thoughtful and constructive comments that greatly improved the quality and presentation of this article.

Additional information

Funding

This research was supported in part by National Science Foundation (NSF) grants CMMI-1233141, CMMI-1362876, DMS-1613258, and DMS-1830344, through the Georgia Institute of Technology, and in part by NSF grant CIF-1513373, through Rutgers University.