Fixed-level calibration of the Cauchy combination test

TL;DR

This paper investigates the fixed-level accuracy of the Cauchy combination test (CCT) under dependence as the number of combined p-values grows, and proposes a boundary-layer calibration method to correct size distortions.

Contribution

It introduces the boundary-layer calibrated CCT (BL-CCT), a new method that corrects size distortions without changing the test statistic, ensuring fixed-level accuracy under dependence.

Findings

Raw CCT is not asymptotically exact at fixed levels under dependence.

BL-CCT achieves asymptotic exactness under weaker dependence conditions.

Numerical experiments confirm the theoretical results.

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 $\alpha\downarrow0$ 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 $\alpha$. 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 $\rho_K$ weakens with $K$, fixed-level behaviour is governed by the boundary-layer scale $s_K=\sqrt{\rho_K}(\log K)^{3/2}$, and the raw CCT is asymptotically exact if and only if $\rho_K(\log K)^3\to0$. 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 $\rho_K\log K\to0$ and provides a useful finite-$K$ approximation on bounded boundary layers. Numerical experiments support the theory.