We study properties of confidence intervals (CIs) for the difference of two Bernoulli distributions' success parameters, $p_x - p_y$, in the case where the goal is to obtain a CI of a given half-width while minimizing sampling costs when the observation costs may be different between the two distributions. Assuming that we are provided with preliminary estimates of the success parameters, we propose three different methods for constructing fixed-width CIs: (i) a two-stage sampling procedure, (ii) a sequential method that carries out sampling in batches, and (iii) an $\ell$-stage "look-ahead" procedure. We use Monte Carlo simulation to show that, under diverse success probability and observation cost scenarios, our proposed algorithms obtain significant cost savings versus their baseline counterparts (up to 50\% for the two-stage procedure, up to 15\% for the sequential methods). Furthermore, for the battery of scenarios under study, our sequential-batches and $\ell$-stage "look-ahead" procedures approximately obtain the nominal coverage while also meeting the desired width requirement. Our sequential-batching method turned out to be more efficient than the "look-ahead" method from a computational standpoint, with average running times at least an order-of-magnitude faster over all the scenarios tested.

中文翻译：

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两个伯努利比例差的经济高效的固定宽度置信区间

我们研究了两个伯努利分布成功参数 $p_x - p_y$ 之间差异的置信区间 (CI) 的属性，其中目标是获得给定半宽的置信区间，同时最小化观察时的抽样成本两种分布的成本可能不同。假设我们获得了成功参数的初步估计，我们提出了三种不同的方法来构建固定宽度的 CI：（i）两阶段抽样程序，（ii）分批抽样的顺序方法，以及（ iii) 一个$\ell$-stage“前瞻”程序。我们使用蒙特卡罗模拟表明，在不同的成功概率和观察成本情景下，我们提出的算法与它们的基线对应物相比获得了显着的成本节约（两阶段程序高达 50%，顺序方法高达 15%）。此外，对于正在研究的一系列场景，我们的顺序批次和 $\ell$ 阶段“前瞻”程序近似地获得了标称覆盖范围，同时也满足了所需的宽度要求。从计算的角度来看，我们的顺序批处理方法比“前瞻”方法更有效，在所有测试的场景中平均运行时间至少快一个数量级。程序近似获得标称覆盖，同时也满足所需的宽度要求。从计算的角度来看，我们的顺序批处理方法比“前瞻”方法更有效，在所有测试的场景中平均运行时间至少快一个数量级。程序近似获得标称覆盖，同时也满足所需的宽度要求。从计算的角度来看，我们的顺序批处理方法比“前瞻”方法更有效，在所有测试的场景中平均运行时间至少快一个数量级。