A Nonparametric Two‐Sample Test Using a Parametric Integral Probability Metric

ABSTRACT

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 asymptotic 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.