标题： Z-Dip：Dip检验的验证推广 显示英文标题

标题： Z-Dip: a validated generalization of the Dip Test

摘要： 在经验分布中检测多模态是统计学和数据分析中的基本问题，其应用范围从聚类到社会科学。哈特igan的Dip检验是一种经典的非参数过程，用于检验单峰性与多模态性，但其解释受到样本量强烈依赖和需要查找表的限制。我们引入了Z-Dip，这是Dip检验的一种标准化扩展，通过将观察到的Dip值与模拟的零分布进行比较，消除了样本量依赖性。我们通过模拟和引导重采样校准了Z-Dip的通用决策阈值，提供了一个统一的多模态检测标准。在最后一节中，我们还提出了一种基于下采样的方法，以进一步减轻非常大数据集中的剩余样本量效应。查找表和软件实现已提供，以便在实践中高效使用。 显示英文摘要

摘要： Detecting multimodality in empirical distributions is a fundamental problem in statistics and data analysis, with applications ranging from clustering to social science. Hartigan's Dip Test is a classical nonparametric procedure for testing unimodality versus multimodality, but its interpretation is hindered by strong dependence on sample size and the need for lookup tables. We introduce the Z-Dip, a standardized extension of the Dip Test that removes sample-size dependence by comparing observed Dip values to simulated null distributions. We calibrate a universal decision threshold for Z-Dip via simulation and bootstrap resampling, providing a unified criterion for multimodality detection. In the final section, we also propose a downsampling-based approach to further mitigate residual sample-size effects in very large datasets. Lookup tables and software implementations are made available for efficient use in practice.