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 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, [], covers the parameter with confidence at least where 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.

中文翻译：

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伯努利比例的串联宽度连续置信区间

摘要 我们提出了一种用于获得伯努利比例 p 的串联宽度置信区间的两阶段顺序方法。术语“串联宽度”是指置信区间的半宽不是预先固定的；它需要满足两个不同的半角上限 h0 和 h1，具体取决于 p 的（未知）值。为了解决这个问题，我们首先提出了一种简单但有用的顺序方法来获得 p 的固定宽度置信区间，其停止规则基于 p 的极大极小估计量。我们依次观察 Bernoulli(p) 试验，对于某些固定的半宽度 h = h0 或 h1，我们开发了一个停止时间 T，使得 p 的结果置信区间 []，至少有信心地覆盖参数p 在时间 T 的最大似然估计量。此外，我们推导出我们提出的固定宽度和串联宽度方法的理论特性，并将它们的性能与现有的替代顺序方案进行比较。所提出的基于极大极小值的固定宽度方法的性能与其他固定宽度方法相似，但在实践中更容易实现。此外，与固定宽度对应方法相比，所提出的串联宽度方法可有效节省样本量，并在没有 p 的先验知识可用时为科学家提供出色的结果。