发表机构
University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对重尾p值聚合检验，提出二阶校准理论，揭示相依性几何决定校准误差的速率、系数与方向，并给出可积传递等关键定理。
AI 中文摘要
在未知相依性下，重尾 $p$ 值组合检验具有吸引力，因为即使精确的相依零分布不可用，其零尾部也能实现一阶稳健性。然而，这种稳健性并不能解决校准问题：$\Pr\{T>q(\alpha)\}=\alpha+o(\alpha)$ 既不能量化剩余的尺寸误差，也不能确定其符号。我们为正向半柯西和倒数（即调和平均）聚合发展了一套二阶校准理论。在精确边际标准化之后，极值相依性由坐标面格上的指数为一的指数测度表示。其支撑集区分了轴向、全内部、真面和混合几何。一个 Möbius 分解展示了非紧加权半空间如何探测每个面，而定量面极限则决定校正是可积的、临界放大的还是不可积的。通过共同因子反演和尾部到校准的映射，这套机制可应用于单个参数 Copula 族之外。结果包括一个可积隐藏面传递定理、一个具有显式高阶面控制的锐利固定维局部角定理、一个共同重因子定理，以及相对于独立性的尺寸和临界值展开。高斯、标准多元 $t$、正 Clayton 和最大线性对冲击模型展现出不同的幂、对数、径向-角向和真面机制；高斯加权半空间传递以显式面边界假设为条件。因此，相依性几何决定了由一阶有效性未解决的校准误差的速率、系数和方向。渐近性使用 $t\to\infty$，等价于显著性水平趋于零。
英文摘要
Heavy-tailed $p$-value combination tests are attractive under unknown dependence because their null tails can be first-order robust even when the exact dependent null distribution is unavailable. That robustness does not resolve calibration: $\Pr\{T>q(α)\}=α+o(α)$ neither quantifies the remaining size error nor determines its sign. We develop a second-order calibration theory for positive Half-Cauchy and reciprocal, or harmonic-mean, aggregation. After exact marginal standardization, extremal dependence is represented by an index-one exponent measure on the coordinate-face lattice. Its support separates axial, full-interior, proper-face, and mixed geometries. A Möbius decomposition shows how the noncompact weighted half-space probes every face, while quantitative face limits determine whether the correction is integrable, critically amplified, or nonintegrable. With common-factor inversion and a tail-to-calibration map, this yields machinery applicable beyond individual parametric copula families. The results include an integrable hidden-face transfer theorem, a sharp fixed-dimensional local-corner theorem with explicit higher-face control, a common-heavy-factor theorem, and size and critical-value expansions relative to independence. Gaussian, standard multivariate-$t$, positive Clayton, and max-linear pair-shock models exhibit distinct power, logarithmic, radial--angular, and proper-face mechanisms; the Gaussian weighted-half-space transfer is conditional on explicit face-boundary hypotheses. Thus dependence geometry determines the rate, coefficient, and direction of the calibration error left unresolved by first-order validity. Asymptotics use $t\to\infty$, equivalently vanishing significance levels.
