The concept of independence plays a crucial role in probability theory and has been the subject of extensive research in recent years. Numerous approaches have been proposed to validate this dependency, but most of them address the problem only at a global level. From a practical perspective, it is important not only to determine whether the data is dependent, but also to identify where this dependence occurs and how strong it is. We introduce a new method for testing statistical independence using the quantile dependence function. Rather than assessing whether the value of the test statistic exceeds a single critical threshold and subsequently deciding whether to reject the independence hypothesis, we use so-called critical surfaces that guarantee locally equal probability of exceeding it under independence. This approach enables a detailed examination of local discrepancies and an assessment of their statistical significance while preserving the overall significance level of the test.

翻译：独立性概念在概率论中扮演着关键角色，近年来已成为广泛研究的主题。虽然已有多种方法被提出以验证这种依赖性，但大多数仅从全局层面处理该问题。从实践角度来看，不仅需要确定数据是否具有依赖性，还需识别这种依赖性发生在何处及其强度如何。本文提出一种基于分位数依赖函数检验统计独立性的新方法。不同于通过检验统计量值是否超过单一临界阈值来决定是否拒绝独立性假设，我们采用所谓的临界曲面方法——该曲面保证在独立条件下具有局部相等的超越概率。此方法能够在保持检验整体显著性水平的前提下，实现对局部差异的精细考察及其统计显著性的评估。