Detection coherence of tests
Abstract
There exist tests calibrated under a null narrower than the one implied by their test statistic -- the detection-null set. The part of the detection-null set not in the null is the test's blind spot. A framework for assessing detection coherence -- whether a test's blind spot is empty -- is introduced, built on new concepts of calibration statistic, detector, detection functional, detection-null set, and blind spot. It is demonstrated that the Wilcoxon--Mann--Whitney, Kruskal--Wallis, Friedman, and logrank tests are detection incoherent as unrestricted tests. Detection incoherent tests may become coherent under restrictions that make the blind spot empty, though domain restriction is not a reliable route to coherence in practice, since the boundary of the coherence-restoring domain is unknown and unverifiable. The Kolmogorov--Smirnov and Zaremba tests, and the Maximum Mean Discrepancy test with a characteristic kernel, are shown to be universally detection coherent. Due to the blind spot, a detection incoherent test misses discoveries when non-rejecting and yields spurious discoveries when rejecting, in both cases regardless of sample size. Detection incoherent tests should be abandoned.
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BibTeX
@misc{grendar2026detection,
title = {{Detection coherence of tests}},
author = {Marian Grendar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.11177},
eprint = {2608.11177},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.11177},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Detection coherence of tests
AU - Marian Grendar
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2608.11177
UR - https://arxiv.org/abs/2608.11177
AB - There exist tests calibrated under a null narrower than the one implied by their test statistic -- the detection-null set. The part of the detection-null set not in the null is the test's blind spot. A framework for assessing detection coherence -- whether a test's blind spot is empty -- is introduced, built on new concepts of calibration statistic, detector, detection functional, detection-null set, and blind spot. It is demonstrated that the Wilcoxon--Mann--Whitney, Kruskal--Wallis, Friedman, and logrank tests are detection incoherent as unrestricted tests. Detection incoherent tests may become coherent under restrictions that make the blind spot empty, though domain restriction is not a reliable route to coherence in practice, since the boundary of the coherence-restoring domain is unknown and unverifiable. The Kolmogorov--Smirnov and Zaremba tests, and the Maximum Mean Discrepancy test with a characteristic kernel, are shown to be universally detection coherent. Due to the blind spot, a detection incoherent test misses discoveries when non-rejecting and yields spurious discoveries when rejecting, in both cases regardless of sample size. Detection incoherent tests should be abandoned.
ER -