Exact grid-free confidence-region computation from dependent p -value functions under arbitrary dependence

Abstract

Repeated randomization workflows, such as sample splitting and conformal ensembling, frequently produce multiple dependent valid p-value functions for the same target. Existing p-merging theory guarantees pointwise validity under arbitrary dependence, but confidence-region inversion typically requires costly numerical gridding. We study when confidence regions can be computed exactly from split-wise regions without gridding the parameter space. We show that mergers induced by step calibrators form a broad class exactly executable at the region level via finite-layer voting, and a partial converse within the calibrator-induced family ties exact executability to step structure. We develop exact voting algorithms, including an adaptive multi-quantile contour aggregator that avoids pre-specifying a single order-statistic threshold while preserving finite-sample validity under arbitrary dependence. Simulations in repeated-split regression and conformal prediction, and real-data regression examples, show substantial computational gains over grid-inversion comparators while eliminating the discretization error introduced by the inversion grid.