Detecting distributional differences between two independent samples is a fundamental problem in statistics and machine learning. Nonparametric two-sample testing provides a principled framework for determining whether two samples are drawn from the same underlying distribution, without assuming any specific parametric form for the distribution. In this study, we propose a new two-sample test statistic based on a newly introduced integral probability metric (IPM), using a specially designed parametric discriminator class with a single node of a neural network. We show that the resulting test statistic, called PReLU-IPM, is nonparametric and establish theoretical guarantees for the associated two-sample testing procedure, PReLU-TST, including its consistency and asymptotical equivalence to nonparametric IPM-based tests under regularity conditions. By analyzing multiple simulated and real benchmark datasets, we demonstrate that PReLU-TST achieves higher power across a range of alternatives or performs comparably to its competitors, for finite samples.

翻译：检测两个独立样本之间的分布差异是统计学和机器学习中的基本问题。非参数双样本检验提供了一个原则性框架，用于确定两个样本是否来自相同的潜在分布，而无需假设任何具体的分布参数形式。在本研究中，我们基于新引入的积分概率度量（IPM），利用一个具有神经网络单节点的专门设计的参数化判别器类，提出了一种新的双样本检验统计量。我们证明，所得到的检验统计量称为PReLU-IPM是非参数的，并为相关的双样本检验程序PReLU-TST建立了理论保证，包括其在正则条件下的相合性以及与基于非参数IPM的检验的渐近等价性。通过分析多个模拟和真实基准数据集，我们证明，对于有限样本，PReLU-TST在一系列备择假设下实现了更高的检验功效，或者与竞争对手表现相当。