Paper
Fixed-level calibration of the Cauchy combination test
Authors
Hirofumi Ota
Abstract
The Cauchy combination test (CCT) is widely used because it gives a closed-form combined $p$-value and is known to be asymptotically valid as the nominal level $α\downarrow0$ under broad dependence structures. We study a different asymptotic question: whether the usual Cauchy cutoff remains accurate at an ordinary fixed level when the number $K$ of combined $p$-values grows under dependence. Under a canonical one-factor equicorrelated Gaussian copula model, we show that the raw CCT is generally not asymptotically exact at fixed $α$. With fixed positive correlation, the statistic converges to a random latent-factor limit, so there is no universal fixed-level reference law. When the common correlation $ρ_K$ weakens with $K$, fixed-level behaviour is governed by the boundary-layer scale $s_K=\sqrt{ρ_K}(\log K)^{3/2}$, and the raw CCT is asymptotically exact if and only if $ρ_K(\log K)^3\to0$. Because the size distortion arises entirely from the reference law and not from the statistic, it can be corrected without modifying the test statistic itself. We propose the boundary-layer calibrated CCT (BL-CCT), which replaces the standard Cauchy reference by a one-parameter Gaussian-smoothed Cauchy family while keeping the statistic unchanged. This reference-law correction is fundamentally different from existing approaches that modify the test statistic. BL-CCT is asymptotically exact under the weaker condition $ρ_K\log K\to0$ and provides a useful finite-$K$ approximation on bounded boundary layers. Numerical experiments support the theory.
Metadata
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Raw Data (Debug)
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"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2603.22668v1</id>\n <title>Fixed-level calibration of the Cauchy combination test</title>\n <updated>2026-03-24T00:42:51Z</updated>\n <link href='https://arxiv.org/abs/2603.22668v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2603.22668v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>The Cauchy combination test (CCT) is widely used because it gives a closed-form combined $p$-value and is known to be asymptotically valid as the nominal level $α\\downarrow0$ under broad dependence structures. We study a different asymptotic question: whether the usual Cauchy cutoff remains accurate at an ordinary fixed level when the number $K$ of combined $p$-values grows under dependence. Under a canonical one-factor equicorrelated Gaussian copula model, we show that the raw CCT is generally not asymptotically exact at fixed $α$. With fixed positive correlation, the statistic converges to a random latent-factor limit, so there is no universal fixed-level reference law. When the common correlation $ρ_K$ weakens with $K$, fixed-level behaviour is governed by the boundary-layer scale $s_K=\\sqrt{ρ_K}(\\log K)^{3/2}$, and the raw CCT is asymptotically exact if and only if $ρ_K(\\log K)^3\\to0$. Because the size distortion arises entirely from the reference law and not from the statistic, it can be corrected without modifying the test statistic itself. We propose the boundary-layer calibrated CCT (BL-CCT), which replaces the standard Cauchy reference by a one-parameter Gaussian-smoothed Cauchy family while keeping the statistic unchanged. This reference-law correction is fundamentally different from existing approaches that modify the test statistic. BL-CCT is asymptotically exact under the weaker condition $ρ_K\\log K\\to0$ and provides a useful finite-$K$ approximation on bounded boundary layers. Numerical experiments support the theory.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='math.ST'/>\n <category scheme='http://arxiv.org/schemas/atom' term='stat.ME'/>\n <published>2026-03-24T00:42:51Z</published>\n <arxiv:primary_category term='math.ST'/>\n <author>\n <name>Hirofumi Ota</name>\n </author>\n </entry>"
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