Abstract Some hypothesis tests for analyzing the degree of overlap between the expected value of random intervals are provided. For this purpose, a suitable measure to quantify the overlapping grade between intervals is considered on the basis of the Szymkiewicz-Simpson coefficient defined for general sets. It can be seen as a kind of likeness index to measure the mutual information between two intervals. On one hand, an estimator for the proposed degree of overlap between intervals is provided and its strong consistency is analyzed. On the other hand, two tests are also proposed in this framework: a one-sample test to examine the degree of overlap between the expected value of a random interval and a given interval, and a two-sample test to check the degree of overlap between the expected value of two random intervals. To solve such hypothesis tests, two statistics are suggested and their limit distributions are studied by considering both asymptotic and bootstrap techniques. Their power has been also explored by means of local alternatives. In addition, some simulation studies are carried out to investigate the behavior of the proposed approaches. Finally, the performance of the tests is also reported in a real-life application.

中文翻译：

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测试随机区间期望值的重叠程度

摘要 提供了一些用于分析随机区间期望值之间重叠程度的假设检验。为此，在为一般集合定义的 Szymkiewicz-Simpson 系数的基础上，考虑了量化区间之间重叠等级的合适度量。可以看作是一种相似度指标，用来衡量两个区间之间的互信息。一方面，提供了建议的区间之间重叠程度的估计量，并分析了其强一致性。另一方面，该框架还提出了两个测试：一个是检查随机区间的期望值与给定区间的重叠程度的单样本测试，以及检查重叠程度的双样本测试两个随机区间的期望值之间。为了解决这样的假设检验，提出了两个统计量，并通过考虑渐近和自举技术来研究它们的极限分布。还通过当地替代品探索了它们的力量。此外，还进行了一些模拟研究以研究所提出方法的行为。最后，还在实际应用中报告了测试的性能。