Testing Equality of Distributions via Repeatedly Integrated Quantile Metrics Under Weak Moment Conditions
Abstract
Testing whether two independent samples arise from the same underlying distribution is a fundamental statistical problem. We propose a new class of two-sample distribution tests based on a family of probability metrics Δn,p, constructed from repeatedly integrated quantile functions. On their respective domains, these metrics are proved to be genuine distributional distances. The case n=1 recovers the p-Wasserstein distance, which requires finite p-th moments; for n≥2, the proposed metrics are well defined and require only finite first moments. The asymptotic properties of the plug-in statistic are established, including strong consistency and limiting distributions under the null and fixed alternatives. A permutation calibration for finite-sample inference is also proposed. We further derive an asymptotic power function under local alternatives. Finally, the finite-sample performance of the proposed tests is examined through simulation studies, and their reduced sensitivity to extreme upper-tail observations is illustrated through a real data application.
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BibTeX
@misc{zou2026testing,
title = {{Testing Equality of Distributions via Repeatedly Integrated Quantile Metrics Under Weak Moment Conditions}},
author = {Zhenfeng Zou and Meng Guan and Panxu Yuan and Sanying Feng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04935},
eprint = {2609.04935},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04935},
}
RIS
TY - UNPB
TI - Testing Equality of Distributions via Repeatedly Integrated Quantile Metrics Under Weak Moment Conditions
AU - Zhenfeng Zou
AU - Meng Guan
AU - Panxu Yuan
AU - Sanying Feng
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.04935
UR - https://arxiv.org/abs/2609.04935
AB - Testing whether two independent samples arise from the same underlying distribution is a fundamental statistical problem. We propose a new class of two-sample distribution tests based on a family of probability metrics Δ_n,p, constructed from repeatedly integrated quantile functions. On their respective domains, these metrics are proved to be genuine distributional distances. The case n=1 recovers the p-Wasserstein distance, which requires finite p-th moments; for n≥2, the proposed metrics are well defined and require only finite first moments. The asymptotic properties of the plug-in statistic are established, including strong consistency and limiting distributions under the null and fixed alternatives. A permutation calibration for finite-sample inference is also proposed. We further derive an asymptotic power function under local alternatives. Finally, the finite-sample performance of the proposed tests is examined through simulation studies, and their reduced sensitivity to extreme upper-tail observations is illustrated through a real data application.
ER -