Title: The Kolmogorov-Smirnov Statistic Revisited Show Chinese title

Title: 重访科尔莫戈罗夫-斯米尔诺夫统计量

Abstract: The Kolmogorov-Smirnov (KS) statistic is a classical nonparametric test widely used for comparing an empirical distribution function with a reference distribution or for comparing two empirical distributions. Despite its broad applicability in statistical hypothesis testing and model validation, certain aspects of the KS statistic remain under-explored among the young generation, particularly under finite sample conditions. This paper revisits the KS statistic in both one-sample and two-sample scenarios, considering one-sided and two-sided variants. We derive exact probabilities for the supremum of the empirical process and present a unified treatment of the KS statistic under diverse settings. Additionally, we explore the discrete nature of the hitting times of the normalized empirical process, providing practical insights into the computation of KS test p-values. The study also discusses the Dvoretzky-Kiefer-Wolfowitz-Massart (DKWM) inequality, highlighting its role in constructing confidence bands for distribution functions. Using empirical process theory, we establish the limit distribution of the KS statistic when the true distribution includes unknown parameters. Our findings extend existing results, offering improved methodologies for statistical analysis and hypothesis testing using the KS statistic, particularly in finite sample scenarios. Show Chinese abstract

Abstract: Kolmogorov-Smirnov（KS）统计量是一种经典的非参数检验，广泛用于比较经验分布函数与参考分布，或者比较两个经验分布。 尽管它在统计假设检验和模型验证中有广泛应用，但关于KS统计量的某些方面在年轻一代中仍缺乏深入探讨，尤其是在有限样本条件下。 本文重新审视了一样本和两样本情况下的KS统计量，并考虑了单侧和双侧变体。 我们推导了经验过程 supremum 的精确概率，并在不同设定下对KS统计量进行了统一处理。 此外，我们探索了标准化经验过程命中时间的离散性质，为计算KS检验的p值提供了实用见解。 研究还讨论了Dvoretzky-Kiefer-Wolfowitz-Massart（DKWM）不等式，强调了其在构造分布函数置信带中的作用。 利用经验过程理论，我们建立了当真实分布包含未知参数时KS统计量的极限分布。 我们的发现扩展了现有结果，为使用KS统计量进行统计分析和假设检验提供了改进的方法，特别是在有限样本场景中。