A nonparametric two-sample test using a parametric integral probability metric

2026-06-15

AI Summary

This study addresses the two-sample testing problem without distributional assumptions. It proposes PReLU-TST, a novel approach based on integral probability metrics (IPMs), which constructs a nonparametric test statistic using a parametrized discriminator consisting of only a single neuron. This design retains the flexibility of nonparametric methods while substantially improving computational efficiency. The proposed test is proven to be consistent and asymptotically equivalent to classical nonparametric IPM-based tests. Empirical evaluations demonstrate that PReLU-TST achieves higher or at least comparable finite-sample testing power against existing methods across a range of synthetic and real-world datasets.

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 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.

Problem

• two-sample testing
• nonparametric test
• distributional difference
• integral probability metric
• statistical consistency

Innovation

• integral probability metric
• nonparametric two-sample test
• neural network discriminator
• statistical consistency
• PReLU-TST