Simultaneous confidence bands for cumulative hazard via exchangeable bootstrap and box calibration

2026-06-29

AI Summary

Existing resampling-based simultaneous confidence bands for the cumulative hazard function often suffer from undercoverage in finite samples with right-censoring. This work proposes a novel approach that integrates an exchangeable bootstrap with box calibration, preserving the ratio structure of the Nelson–Aalen estimator and constructing upper and lower step-function envelopes over a linearly scanned grid of event times. By doing so, it effectively calibrates vertical deviations without requiring variance-stabilizing transformations. The method substantially improves coverage accuracy in finite samples, achieving empirical coverage probabilities much closer to the nominal level across diverse hazard function shapes and censoring rates. Its practical advantage is further demonstrated through an application to melanoma data, where it yields visibly refined simultaneous confidence bands for the cumulative hazard.

Abstract

Resampling-based simultaneous confidence bands for cumulative hazard functions often undercover in finite samples with right censoring. We study two aspects of the construction that can contribute to this gap, the resampling scheme and the calibration statistic, and propose a procedure that intervenes on both. The exchangeable bootstrap reweights the numerator and the denominator of the Nelson-Aalen ratio, preserving its ratio structure. The box-calibrated discrepancy constructs lower and upper step envelopes from adjacent values of the original and resampled Nelson-Aalen estimators and measures the resulting vertical discrepancy. We establish conditional weak convergence of the exchangeable bootstrap, prove that box calibration is first-order asymptotically equivalent to grid calibration, and show that the resulting band attains nominal coverage asymptotically. The box correction uses the same bootstrap paths and event-time grid as grid calibration; after each bootstrap path is formed, it requires only an additional linear pass over the event-time grid and therefore has negligible computational overhead. In simulations across a range of hazard shapes and censoring levels, the exchangeable bootstrap with box calibration is, in most configurations, closest to nominal coverage among the methods considered. A notable consequence is a ranking reversal: the ratio-preserving exchangeable bootstrap has the lowest coverage under grid calibration, yet is usually closest to the nominal level after box calibration. A melanoma data example illustrates the practical effect on the cumulative hazard bands. The proposed procedure operates on the original cumulative-hazard scale, requires no variance-stabilizing transformation, and permits inference from time zero.

Problem

• simultaneous confidence bands
• cumulative hazard
• right censoring
• coverage probability
• resampling

Innovation

• exchangeable bootstrap
• box calibration
• simultaneous confidence bands
• cumulative hazard
• Nelson-Aalen estimator