## 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 - α)% confidence interval is not fixed beforehand; it is instead required to satisfy two different half-width upper bounds, h_{0} and h_{1}, 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 = h_{0} or h_{1}, 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.

Original language | English (US) |
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Pages (from-to) | 163-183 |

Number of pages | 21 |

Journal | Sequential Analysis |

Volume | 38 |

Issue number | 2 |

DOIs | |

State | Published - Apr 3 2019 |

## Keywords

- 60G40
- 62L12
- Bernoulli proportion
- binomial distribution
- confidence interval
- sequential analysis
- stopping time

## ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation