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arxiv: 2606.30982 · v1 · pith:XY3AGBRDnew · submitted 2026-06-29 · 📊 stat.ME

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

Min Lin , Grzegorz Rempala , Eben Kenah , Qianying Lin This is my paper

Pith reviewed 2026-07-01 00:53 UTC · model grok-4.3

classification 📊 stat.ME
keywords simultaneous confidence bandscumulative hazardNelson-Aalen estimatorexchangeable bootstrapbox calibrationright censoringresampling
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The pith

Exchangeable bootstrap with box calibration yields simultaneous confidence bands for cumulative hazard that attain nominal asymptotic coverage.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Resampling-based simultaneous confidence bands for cumulative hazard functions often undercover in finite samples with right censoring. The paper intervenes on both the resampling scheme and the calibration statistic. It introduces an exchangeable bootstrap that reweights the numerator and denominator of the Nelson-Aalen ratio to preserve its structure, and a box calibration that measures vertical discrepancy via lower and upper step envelopes from adjacent estimator values. The resulting band attains nominal coverage asymptotically while adding negligible computational cost and operating directly on the original scale from time zero.

Core claim

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.

What carries the argument

Exchangeable bootstrap that reweights both numerator and denominator of the Nelson-Aalen ratio estimator while preserving its ratio structure, paired with box calibration that constructs step envelopes from adjacent values to measure vertical discrepancy.

Load-bearing premise

The exchangeable bootstrap satisfies the conditions for conditional weak convergence to the limiting Gaussian process.

What would settle it

Large-sample simulations in which empirical coverage probabilities fail to approach the nominal level would falsify the asymptotic coverage claim.

Figures

Figures reproduced from arXiv: 2606.30982 by Eben Kenah, Grzegorz Rempala, Min Lin, Qianying Lin.

Figure 1
Figure 1. Figure 1: Illustration of the box calibration on a sample with uncensored event times [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Scaled signed coverage deviation √ 𝑛{covg𝑛(𝛼) − (1 − 𝛼)} against sample size 𝑛 on a logarithmic scale, for the three box-calibrated methods under heavy censoring (∼ 50%). Rows correspond to the four hazard shapes; columns correspond to the three bootstrap schemes. Point shape and color encode the nominal level 𝛼 ∈ {0.01, 0.05, 0.10}. The grey ribbon marks the 95% Monte Carlo reference band ±1.96√︁ 𝑛𝛼(1 − 𝛼… view at source ↗
Figure 3
Figure 3. Figure 3: Melanoma data from Andersen et al. (1993), restricted to the 79 male patients. The solid black step function is the Nelson–Aalen estimator for the cumulative hazard of death from melanoma. The solid blue step functions show XBoot–Box on [0, 7], the dotted red step functions show HW on [0, 7], and the dot-dashed red step functions show logHW on [1, 7]. The logHW band is included to mirror the classical pres… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes an exchangeable bootstrap for the Nelson-Aalen estimator that reweights both the numerator and denominator to preserve the ratio structure, paired with a box-calibrated discrepancy that constructs lower and upper step envelopes from adjacent values of the original and resampled estimators. It claims to establish conditional weak convergence of this bootstrap process, prove that box calibration is first-order asymptotically equivalent to grid calibration, and show that the resulting simultaneous confidence bands attain nominal coverage asymptotically. Simulations across hazard shapes and censoring levels indicate the method is often closest to nominal coverage, with a ranking reversal relative to grid calibration, and the approach is illustrated on a melanoma data example.

Significance. If the stated results hold, the work provides a computationally light method (only an additional linear pass over the event-time grid) for simultaneous bands on the cumulative-hazard scale that avoids variance-stabilizing transformations. The proofs of conditional weak convergence, asymptotic equivalence, and coverage, together with the simulation comparisons, constitute concrete strengths that could improve finite-sample performance in right-censored survival analysis.

major comments (2)
  1. [Abstract] Abstract: the claim that conditional weak convergence of the exchangeable bootstrap is established is load-bearing for the coverage result, yet the abstract supplies no explicit regularity conditions on the censoring survival function (e.g., positivity and continuity on the support of the event times) or on the cumulative hazard itself; without these, the scope of the convergence under standard right-censoring regimes cannot be verified.
  2. [Abstract] Abstract: the proof that the resulting band attains nominal coverage asymptotically relies on the box calibration being first-order equivalent to grid calibration, but the abstract does not indicate where the interaction between the ratio-preserving reweighting and the step-envelope discrepancy is shown to preserve the necessary tightness and finite-dimensional convergence.
minor comments (1)
  1. The abstract refers to 'a notable consequence is a ranking reversal'; a parenthetical reference to the relevant simulation table or figure would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments on the abstract. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that conditional weak convergence of the exchangeable bootstrap is established is load-bearing for the coverage result, yet the abstract supplies no explicit regularity conditions on the censoring survival function (e.g., positivity and continuity on the support of the event times) or on the cumulative hazard itself; without these, the scope of the convergence under standard right-censoring regimes cannot be verified.

    Authors: The regularity conditions are stated explicitly in Assumption 1 (Section 2 of the manuscript), which requires positivity and continuity of the censoring survival function on the support of the observed event times together with boundedness of the cumulative hazard. These are the standard conditions under which the Nelson-Aalen estimator is consistent and asymptotically Gaussian in right-censored data. While the abstract must remain concise, we agree that a brief reference to the conditions would better indicate the scope. We will revise the abstract to add the clause 'under standard regularity conditions on the censoring distribution and cumulative hazard'. revision: yes

  2. Referee: [Abstract] Abstract: the proof that the resulting band attains nominal coverage asymptotically relies on the box calibration being first-order equivalent to grid calibration, but the abstract does not indicate where the interaction between the ratio-preserving reweighting and the step-envelope discrepancy is shown to preserve the necessary tightness and finite-dimensional convergence.

    Authors: The conditional weak convergence of the exchangeable bootstrap (which incorporates the ratio-preserving reweighting) is proved in Theorem 2. The first-order asymptotic equivalence of box calibration to grid calibration, including verification that the step-envelope discrepancy preserves tightness and finite-dimensional convergence, is established in Theorem 4. The nominal asymptotic coverage of the resulting band then follows in Theorem 5. We will revise the abstract to reference these results, for example by appending 'as shown in Theorems 2--5'. revision: yes

Circularity Check

0 steps flagged

No circularity: asymptotic coverage derived from independent convergence proofs

full rationale

The paper establishes conditional weak convergence of the exchangeable bootstrap, proves first-order asymptotic equivalence between box and grid calibration, and shows nominal coverage from those results. These steps are mathematical derivations on independently defined objects (reweighting scheme, discrepancy measure) rather than reductions of the coverage probability to a fitted parameter or self-citation chain by construction. No quoted equation or definition equates the target coverage to an input quantity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides no explicit free parameters or invented entities. The work rests on standard domain assumptions for the Nelson-Aalen estimator.

axioms (1)
  • domain assumption Observations are subject to independent right censoring
    Required for validity of the Nelson-Aalen estimator and the stated bootstrap convergence.

pith-pipeline@v0.9.1-grok · 5803 in / 1158 out tokens · 54508 ms · 2026-07-01T00:53:02.737867+00:00 · methodology

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Reference graph

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