Copula Geometry and Second-Order Calibration of Heavily Right Aggregation
Abstract
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→∞, equivalently vanishing significance levels.
Related
- Aggregation Distortion in Multilevel Mediation: Estimand Geometry, Design Dependence, and Diagnostics
- Helstrom geometry with time-dependent spectral calibration
- Posterior Geometry and Identifiability in Multi-Response Bayesian Calibration
- When Unpaired Sets Support Shared-Corruption Calibration: Moment Geometry and Two-Sample Precision
- Exact Calibration and Sharp Risk Geometry for Volume-Sampled Ridge Regression
BibTeX
@misc{liu2026copula,
title = {{Copula Geometry and Second-Order Calibration of Heavily Right Aggregation}},
author = {Tianle Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29658},
eprint = {2609.29658},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29658},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Copula Geometry and Second-Order Calibration of Heavily Right Aggregation
AU - Tianle Liu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.29658
UR - https://arxiv.org/abs/2609.29658
AB - 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→∞, equivalently vanishing significance levels.
ER -