A Heavily Right Strategy for Statistical Inference with Dependent Studies in Any Dimension

TL;DR

This work introduces a heavily-right strategy for inference with dependent studies by inverting combination tests built from heavy-tailed distributions, notably the Half-Cauchy Combination Test (HCCT) and Exact Harmonic Mean P-value (EHMP). By truncating the left tail of the Cauchy, HCCT yields convex, bounded confidence regions under common summarizations (Hotelling $T^2$ or $\,oldsymbol{hi}^{2}$), enabling exact computation even with unknown dependencies. Extending to arbitrary dimensions via divide-and-combine projections, the authors provide a scalable framework that avoids full covariance estimation while delivering simultaneous confidence regions and intervals, demonstrated in a network meta-analysis setting. Theoretical results anchor the methods in extreme-value theory (Landau domain of attraction) and tail independence, with practical calibration, adaptive handling of empty sets, and open problems outlined for future work. The approach offers a robust, dependence-agnostic alternative to classical methods, with broad applicability to high-dimensional inference and meta-analytic contexts.

Abstract

We leverage recent advances in heavy-tail approximations for global hypothesis testing with dependent studies to construct approximate confidence regions without modeling or estimating their dependence structures. A non-rejection region is a confidence region but it may not be convex. Convexity is appealing because it ensures any one-dimensional linear projection of the region is a confidence interval, easy to compute and interpret. We show why convexity fails for nearly all heavy-tail combination tests proposed in recent years, including the influential Cauchy combination test. These insights motivate a \textit{heavily right} strategy: truncating the left half of the Cauchy distribution to obtain the Half-Cauchy combination test. The harmonic mean test also corresponds to a heavily right distribution with a Cauchy-like tail, namely a Pareto distribution with unit power. We prove that both approaches guarantee convexity when individual studies are summarized by Hotelling $T^2$ or $χ^{2}$ statistics (regardless of the validity of this summary) and provide efficient, \textit{exact} algorithms for implementation. Applying these methods, we develop a divide-and-combine strategy for mean estimation in any dimension and construct simultaneous confidence intervals in a network meta-analysis for treatment effect comparisons across multiple clinical trials. We also present many open problems and conclude with epistemic reflections.

Figures (19)

• Figure 1: Connectivity of confidence regions for CCT and HCCT.
• Figure 2: Plots of $g_j(\theta) = F_{\nu}^{-1} \bigl\{2F^{(j)} (\lvert \theta \rvert ) - 1 \bigr\}$, where the first distribution in the caption refers to $F_v$, and the second to $F^{(j)}$.
• Figure 3: Confidence intervals from $1$-dimensional HCCT.
• Figure 4: Illustration of obtaining simultaneous confidence intervals from confidence regions via projection. The plot shows $95\%$ and $99\%$ simultaneous confidence intervals for $\boldsymbol{b}_{i}^{\top}\boldsymbol{\theta}$ ($i=1,2$ with $\lVert \boldsymbol{b}_{i} \rVert_{2}=1$).
• Figure 5: Contour plots of confidence regions from $2$-dimensional HCCT.

Theorems & Definitions (36)

• Theorem 2.1
• Theorem 2.2
• Proposition 2.3
• Theorem 5.1
• Proposition 5.2
• Theorem 5.3
• Theorem 5.4
• Corollary 5.5
• Proposition 5.6
• Lemma A.1