Fixed-level calibration of the Cauchy combination test
The Cauchy combination test (CCT) is widely used because it yields 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. Unlike recent variants that modify the test statistic, BL-CCT leaves the statistic unchanged and corrects only the reference law. 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. We also conduct several power analyses: although BL-CCT only raises the cutoff, it incurs no first-order power loss relative to the raw CCT on the exactness scale, under local dense, sparse, and dense Gaussian alternatives. Numerical experiments support the calibration theory.