To detect a change in the probability of a sequence of independent binomial random variables, a variety of asymptotic and exact testing procedures have been proposed. Whenever the sample size or the event rate is small, asymptotic approximations of maximally selected test statistics have been shown to be inaccurate. Although exact methods control the type I error rate, they can be overly conservative due to the discreteness of the test statistics in these situations. We extend approaches by Worsley and Halpern to develop a test that is less discrete to increase the power. Building on ideas from binary segmentation, the proposed test utilizes unused information in the binomial sequences to add a new ordering to test statistics that are of equal value. The exact distributions are derived under side conditions that arise in hypothetical segmentation steps and do not depend on the type of test statistic used (e.g., log likelihood ratio, cumulative sum, or Fisher's exact test). Using the proposed exact segmentation procedure, we construct a change point test and prove that it controls the type-I-error rate at any given nominal level. Furthermore, we prove that the new test is uniformly at least as powerful as Worsley's exact test. In a Monte Carlo simulation study, the gain in power can be remarkable, especially in scenarios with small sample size. Giving a clinical database example about pin site infections and an example assessing publication bias in neuropsychiatric drug research, we demonstrate the wide-ranging applicability of the test.

中文翻译：

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小样本二项式序列中具有改进功效的精确变化点检测

为了检测一系列独立二项式随机变量的概率变化，已经提出了各种渐近和精确的测试程序。每当样本大小或事件率很小时，最大选择的测试统计量的渐近近似已被证明是不准确的。尽管精确方法控制了 I 类错误率，但由于在这些情况下测试统计数据的离散性，它们可能过于保守。我们扩展了 Worsley 和 Halpern 的方法来开发一种不那么离散的测试来增加功效。基于二元分割的思想，建议的测试利用二项式序列中未使用的信息来添加新的排序来测试具有相同价值的统计数据。精确分布是在假设分割步骤中出现的辅助条件下导出的，不依赖于所使用的检验统计量的类型（例如，对数似然比、累积总和或 Fisher 精确检验）。使用所提出的精确分割程序，我们构建了一个变化点测试并证明它在任何给定的标称水平上控制了 I 类错误率。此外，我们证明新的检验至少与 Worsley 的精确检验一样强大。在 Monte Carlo 模拟研究中，功效的增益可能非常显着，尤其是在样本量较小的情况下。给出一个关于针点感染的临床数据库示例和一个评估神经精神药物研究发表偏倚的示例，我们证明了该测试的广泛适用性。