A Heavily Right Strategy for Integrating Dependent Studies in Any Dimensions

Recently, there has been a surge of interest in hypothesis testing methods that can combine dependent studies without addressing their dependence. Among these, the Cauchy Combination Test (CCT) stands out for its approximate validity and power across diverse dependencies, leveraging a heavy-tailed approximation insensitive to tail dependence. However, inverting CCT to construct confidence regions can yield regions lacking compactness, convexity, or connectivity. We propose a “heavily right” strategy by excluding the left half of the Cauchy distribution in the combination rule, retaining CCT’s tail insensitivity while mitigating sensitivity to large $p$-values. We prove that the Half-Cauchy combination produces convex, compact confidence regions, shown theoretically and empirically to be the one of the only “heavily right” heavy-tail distributions with these desirable properties. On the computational side, we introduce efficient algorithms for both methods. For applications, we develop a divide-and-conquer approach for high-dimensional mean estimation using the Half-Cauchy method, which is guaranteed to provide convex and compact confidence regions even when sample sizes are smaller than the parameter dimension. In network meta-analysis, we demonstrate its practical utility in constructing simultaneous confidence intervals for treatment effects in clinical trials.