A Partial-Matching Proof of the Neyman--Pearson Lemma
Abstract
Testing seeks to distinguish distributions, whereas matching seeks to identify their common mass. I show that this apparent opposition provides a direct proof of the Neyman--Pearson lemma. Maximal partial matching between the alternative distribution and a scaled null distribution separates them into a common component and two mutually singular residuals. An elementary dual problem then turns this separation into a testing rule: complementary slackness forces rejection on the alternative residual and non-rejection on the scaled-null residual, leaving freedom to randomize at the likelihood-ratio threshold. Thus the optimal test emerges from maximal matching rather than being proposed in advance. The matching profile also determines the optimal power envelope.
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BibTeX
@misc{tabri2026partialmatching,
title = {{A Partial-Matching Proof of the Neyman--Pearson Lemma}},
author = {Rami V. Tabri},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07368},
eprint = {2611.07368},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07368},
}
RIS
TY - UNPB
TI - A Partial-Matching Proof of the Neyman--Pearson Lemma
AU - Rami V. Tabri
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07368
UR - https://arxiv.org/abs/2611.07368
AB - Testing seeks to distinguish distributions, whereas matching seeks to identify their common mass. I show that this apparent opposition provides a direct proof of the Neyman--Pearson lemma. Maximal partial matching between the alternative distribution and a scaled null distribution separates them into a common component and two mutually singular residuals. An elementary dual problem then turns this separation into a testing rule: complementary slackness forces rejection on the alternative residual and non-rejection on the scaled-null residual, leaving freedom to randomize at the likelihood-ratio threshold. Thus the optimal test emerges from maximal matching rather than being proposed in advance. The matching profile also determines the optimal power envelope.
ER -