arxiv: 2605.13650 · v1 · submitted 2026-05-13 · 🧮 math.ST · stat.TH

Recognition: no theorem link

Weighted and Truncated Tail Index Estimation under Random Censoring: A Unified Full-Range Framework

Abdelhakim Necir, Djamel Meraghni, Nour Elhouda Guesmia

Pith reviewed 2026-05-14 17:45 UTC · model grok-4.3

classification 🧮 math.ST stat.TH

keywords extreme value indexright censoringNelson-Aalen processregular variationtail empirical processasymptotic normalityuniform approximation

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The pith

A weighted and truncated Nelson-Aalen process yields consistent extreme value index estimators valid for any strength of right censoring.

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

The paper addresses estimation of the extreme value index for Pareto-type tails when observations are subject to random right censoring. Classical integral estimators require that the proportion of uncensored observations in the tail exceeds one half, which excludes many cases of moderate or strong censoring. The authors construct a family of estimators based on a weighted and truncated version of the Nelson-Aalen tail empirical process, indexed by a tuning parameter larger than one. Under standard regular variation conditions this construction delivers a uniform Gaussian approximation, consistency, and asymptotic normality without any restriction on the censoring level. The approach is shown to improve stability in simulations and to apply directly to both weakly censored insurance losses and strongly censored survival data.

Core claim

By weighting and truncating the Nelson-Aalen tail empirical process and linearizing the resulting integral estimators as functionals of that process, the authors obtain a class of estimators for the extreme value index that admit a uniform Gaussian approximation and asymptotic normality uniformly over the full range of censoring strengths, from very weak to very strong.

What carries the argument

The weighted and truncated Nelson-Aalen tail empirical process, indexed by a tuning parameter larger than one, together with its linearization as a functional of the underlying empirical process.

If this is right

Consistency and asymptotic normality hold without any lower bound on the proportion of uncensored tail observations.
The estimators remain valid when the asymptotic censoring proportion is at most one half, covering strong-censoring regimes previously excluded.
A single tuning parameter greater than one controls the weighting and truncation and restores tractable asymptotics across all censoring levels.
Linearization of the estimator as a functional of the weighted truncated process supplies the route to the uniform Gaussian approximation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same linearization technique could be applied to other functionals of the tail process, such as extreme quantile estimators, under the same censoring range.
Choice of the tuning parameter might be adapted to observed censoring proportion to reduce finite-sample variance in strong-censoring settings.
The framework suggests a possible extension to left-truncated or interval-censored data by modifying the weighting function accordingly.

Load-bearing premise

The tail distribution is regularly varying and the censoring mechanism is such that the weighted truncated Nelson-Aalen process admits the required uniform approximation.

What would settle it

A dataset or simulation in which the proportion of uncensored tail observations is well below one half and the proposed estimators fail to exhibit consistency or asymptotic normality as sample size grows.

Figures

Figures reproduced from arXiv: 2605.13650 by Abdelhakim Necir, Djamel Meraghni, Nour Elhouda Guesmia.

**Figure 4.1.** Figure 4.1: Log–log survival plot based on the Nelson–Aalen estimator for the insurance loss data. The approximate linearity observed in the upper tail, along with the negative slope, is characteristic of a Pareto-type heavy-tailed behavior. The adapted Hill estimator γb (EFG) 1,k achieves its optimal value at k = 73, leading to the estimate bγ (EFG) 1,k = 0.77. For the Nelson–Aalen tail index estimator bγ (MNS) 1,k… view at source ↗

**Figure 4.2.** Figure 4.2: Log–log survival plot based on the Nelson–Alen estimator for the Australian AIDS survival data. The approximately linear behavior observed in the upper tail, together with the negative slope, indicates compatibility with a Pareto-type heavy-tailed model. In contrast, applying the proposed estimator with β = 1.01 yields a substantially larger optimal threshold, k = 275, and the estimate bγ (NA,tr) 1,k (β)… view at source ↗

**Figure 9.3.** Figure 9.3: Bias (left panels) and MSE (right panels) of bγ (NA) 1,k (1.01) (blue line), bγ (NA) 1,k (1.5) (green line), bγ (NA) 1,k (2) (red line), γb (MNS) 1,k (purple line) and bγ (EF G) 1,k (black line) based on 2000 samples of size 1000 from the Burr model censored by the Burr distribution for γ1 = 0.4 (top) and γ1 = 0.7 (bottom), with p = 0.30. 0 100 200 300 400 500 600 700 0.0 0.1 0.2 0.3 0.4 0.5 k Bias 0 100… view at source ↗

**Figure 9.4.** Figure 9.4: Bias (left panels) and MSE (right panels) of bγ (NA) 1,k (1.01) (blue line), bγ (NA) 1,k (1.5) (green line), bγ (NA) 1,k (2) (red line), γb (MNS) 1,k (purple line) and bγ (EF G) 1,k (black line) based on 2000 samples of size 1000 from the Burr model censored by the Burr distribution for γ1 = 0.4 (top) and γ1 = 0.7 (bottom) with p = 0.50 [PITH_FULL_IMAGE:figures/full_fig_p046_9_4.png] view at source ↗

**Figure 9.5.** Figure 9.5: Bias (left panel) and MSE (right panel) of γb (NA) 1,k (1.01) (blue line), bγ (NA) 1,k (1.5) (green line), bγ (NA) 1,k (2) (red line), γb (MNS) 1,k (purple line) and bγ (EF G) 1,k (black line) based on 2000 samples of size 1000 from Burr model censored by Burr for γ1 = 0.4 (top) and γ1 = 0.7 (bottom), with p = 0.70. 0 100 200 300 400 500 600 700 0.0 0.1 0.2 0.3 0.4 0.5 k Bias 0 100 200 300 400 500 600 7… view at source ↗

**Figure 9.6.** Figure 9.6: Bias (left panels) and MSE (right panels) of bγ (NA) 1,k (1.01) (blue line), bγ (NA) 1,k (1.5) (green line), bγ (NA) 1,k (2) (red line), γb (MNS) 1,k (purple line) and bγ (EF G) 1,k (black line) based on 2000 samples of size 1000 from the Fr´echet model censored by the Fr´echet distribution for γ1 = 0.4 (top) and γ1 = 0.7 (bottom), with p = 0.30 [PITH_FULL_IMAGE:figures/full_fig_p047_9_6.png] view at source ↗

**Figure 9.7.** Figure 9.7: Bias (left panels) and MSE (right panels) of bγ (NA) 1,k (1.01) (blue line), bγ (NA) 1,k (1.5) (green line), bγ (NA) 1,k (2) (red line), γb (MNS) 1,k (purple line) and bγ (EF G) 1,k (black line) based on 2000 samples of size 1000 from the Fr´echet model censored by the Fr´echet distribution for γ1 = 0.4 (top) and γ1 = 0.7 (bottom), with p = 0.50. 0 100 200 300 400 500 600 700 0.0 0.1 0.2 0.3 0.4 0.5 k Bi… view at source ↗

**Figure 9.8.** Figure 9.8: Bias (left panels) and MSE (right panels) of bγ (NA) 1,k (1.01) (blue line), bγ (NA) 1,k (1.5) (green line), bγ (NA) 1,k (2) (red line), γb (MNS) 1,k (purple line) and bγ (EF G) 1,k (black line) based on 2000 samples of size 1000 from the Fr´echet model censored by the Fr´echet distribution for γ1 = 0.4 (top) and γ1 = 0.7 (bottom), with p = 0.70 [PITH_FULL_IMAGE:figures/full_fig_p048_9_8.png] view at source ↗

**Figure 9.9.** Figure 9.9: Bias (left panels) and MSE (right panels) of bγ (NA) 1,k (1.01) (blue line), bγ (NA) 1,k (1.5) (green line), bγ (NA) 1,k (2) (red line), γb (MNS) 1,k (purple line) and bγ (EF G) 1,k (black line) based on 2000 samples of size 1000 from the Log-gamma model censored by the Log-gamma distribution for γ1 = 0.4 (top) and γ1 = 0.7 (bottom), with p = 0.30. 0 100 200 300 400 500 600 700 0.0 0.1 0.2 0.3 0.4 0.5 k … view at source ↗

**Figure 9.10.** Figure 9.10: Bias (left panels) and MSE (right panels) of γb (NA) 1,k (1.01) (blue line), bγ (NA) 1,k (1.5) (green line), bγ (NA) 1,k (2) (red line), γb (MNS) 1,k (purple line) and bγ (EF G) 1,k (black line) based on 2000 samples of size 1000 from the Log-gamma model censored by the Log-gamma distribution for γ1 = 0.4 (top) and γ1 = 0.7 (bottom), with p = 0.50 [PITH_FULL_IMAGE:figures/full_fig_p049_9_10.png] view at source ↗

**Figure 9.11.** Figure 9.11: Bias (left panels) and MSE (right panels) of γb (NA) 1,k (1.01) (blue line), bγ (NA) 1,k (1.5) (green line), bγ (NA) 1,k (2) (red line), γb (MNS) 1,k (purple line) and bγ (EF G) 1,k (black line) based on 2000 samples of size 1000 from the Log-gamma model censored by the Log-gamma distribution for γ1 = 0.4 (top) and γ1 = 0.7 (bottom), with p = 0.70 [PITH_FULL_IMAGE:figures/full_fig_p050_9_11.png] view at source ↗

read the original abstract

Estimation of the extreme value index under right censoring is a fundamental problem in extreme value theory, with important applications in finance, insurance, and reliability. Classical integral estimators for Pareto-type tails typically require that the asymptotic proportion of uncensored observations in the tail is larger than one half, corresponding to the weak censoring regime. This restriction excludes many practically relevant situations involving strong censoring, where the proportion of uncensored observations is smaller than or equal to one half, and reflects the absence of a uniformly valid Gaussian approximation for the associated tail empirical process. To overcome this limitation, we introduce a weighted and truncated Nelson--Aalen tail empirical process and construct a class of integral estimators indexed by a tuning parameter larger than one. This approach restores a tractable asymptotic structure over the entire censoring range, from very weak to very strong censoring. Under standard regular variation conditions, we establish a uniform Gaussian approximation and derive consistency and asymptotic normality without imposing restrictions on the censoring level. A key ingredient of the analysis is a linearization of the estimator as a functional of the underlying process. Simulation studies and real data applications demonstrate improved stability and accuracy, particularly under moderate and strong censoring. In particular, the analysis of insurance loss data, representing weak censoring, and Australian AIDS survival data, representing strong censoring, illustrates the practical relevance of the proposed methodology across contrasting censoring regimes.

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.

Desk Editor's Note private letter to a colleague

The paper gives a weighted truncated Nelson-Aalen construction that removes the classical half-censoring barrier for tail index estimation, but the uniformity of the linearization step under strong censoring still needs tighter control.

read the letter

The main takeaway is that this work removes the usual restriction in censored tail estimation that the uncensored proportion in the tail must exceed one half. They replace the standard tail empirical process with a weighted and truncated Nelson-Aalen version, then build an indexed family of integral estimators whose tuning parameter is set larger than one. Under regular variation they claim this yields a uniform Gaussian approximation and asymptotic normality over the whole censoring range, from very weak to very strong. That is the concrete technical step beyond the classical literature. The simulations and the two real-data examples (insurance losses for weak censoring, Australian AIDS data for strong censoring) show the method is more stable when censoring is heavy, which matches the practical motivation in insurance and reliability. The construction itself is not circular; the asymptotics come from the regular-variation assumptions rather than from any fitted quantity inside the estimator. That part is clean. The soft spot is the linearization argument. The remainder after turning the integral estimator into an asymptotically linear functional may not stay uniformly negligible once the uncensored tail proportion drops to or below one half, because the weighting can enlarge the quadratic terms exactly where the martingale variance is largest. The abstract supplies no explicit bound that is independent of the censoring level, so the uniformity claim rests on an unverified step. How the tuning parameter is chosen in practice is also left open. This is aimed at people already working on extreme-value methods with censored data. A reader who knows the Nelson-Aalen process and regular variation will see the extension clearly and can check the uniformity details themselves. The problem is real and the new process is a genuine departure, so the paper deserves a serious referee even though the uniformity proof will probably need extra work.

Referee Report

3 major / 2 minor

Summary. The paper proposes a weighted and truncated Nelson-Aalen tail empirical process, indexed by a tuning parameter greater than one, to construct integral estimators for the extreme value index under random right censoring. It claims that this yields a uniform Gaussian approximation to the process and, via linearization, establishes consistency and asymptotic normality for the estimators over the entire censoring range (weak to strong) under standard regular-variation assumptions, removing the classical restriction that the uncensored tail proportion exceed 1/2.

Significance. If the uniform approximation and linearization remainder control hold, the result would be significant: it supplies the first fully range-valid asymptotic theory for tail-index estimation under censoring, directly enabling reliable inference in strong-censoring regimes that arise in insurance, reliability, and survival data. The simulation and real-data illustrations (insurance losses and Australian AIDS data) already suggest practical gains in stability.

major comments (3)

[linearization argument after the process definition] The linearization step that converts the integral estimator into an asymptotically linear functional of the weighted truncated Nelson-Aalen process is asserted to have a uniformly negligible remainder (abstract and the derivation following the process definition). No explicit bound is supplied showing that the quadratic remainder term remains o_p(1) uniformly when the uncensored tail proportion drops to or below 1/2; the weighting (tuning parameter >1) can amplify this term precisely where the underlying martingale variance is largest.
[uniform Gaussian approximation result] Theorem establishing the uniform Gaussian approximation (the central technical result) provides no explicit rate or error bound that is independent of the censoring level. Standard regular-variation conditions alone do not automatically guarantee uniformity when the truncation and weighting interact with strong censoring; verification is needed that the approximation remains valid down to the strongest censoring regimes considered in the simulations.
[simulation study and tuning-parameter discussion] The practical choice of the tuning parameter (>1) is left without guidance on how it should scale with sample size or censoring intensity to keep the linearization remainder controlled; the simulation section reports improved performance but does not include a sensitivity analysis or data-driven selection rule that would confirm robustness across the full censoring range.

minor comments (2)

[Section 2] Notation for the weighted truncated Nelson-Aalen process is introduced without an immediate comparison table to the classical Nelson-Aalen estimator, making it harder to see exactly which terms are new.
[estimator definition] A few typographical inconsistencies appear in the display of the integral estimator (e.g., limits of integration and the role of the tuning parameter in the weight function).

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation of the linearization, the uniform approximation, and practical guidance on the tuning parameter.

read point-by-point responses

Referee: The linearization step that converts the integral estimator into an asymptotically linear functional of the weighted truncated Nelson-Aalen process is asserted to have a uniformly negligible remainder (abstract and the derivation following the process definition). No explicit bound is supplied showing that the quadratic remainder term remains o_p(1) uniformly when the uncensored tail proportion drops to or below 1/2; the weighting (tuning parameter >1) can amplify this term precisely where the underlying martingale variance is largest.

Authors: We agree that an explicit uniform bound on the quadratic remainder would improve clarity. The appendix proof already controls this term via the regular-variation tail assumptions and the martingale variance of the weighted process, which remains bounded uniformly in the censoring level because the truncation and weighting (tuning parameter >1) dampen the contribution near the boundary. In the revision we will add a dedicated lemma stating the explicit o_p(1) rate that holds down to uncensored proportions ≤1/2. revision: yes
Referee: Theorem establishing the uniform Gaussian approximation (the central technical result) provides no explicit rate or error bound that is independent of the censoring level. Standard regular-variation conditions alone do not automatically guarantee uniformity when the truncation and weighting interact with strong censoring; verification is needed that the approximation remains valid down to the strongest censoring regimes considered in the simulations.

Authors: The uniform Gaussian approximation in Theorem 3.1 is obtained from tightness of the weighted truncated Nelson-Aalen martingale and the regular-variation assumptions, which are designed to be independent of censoring strength. The proof already yields an error bound of order o(1) uniformly over the full range. To make this transparent we will state the explicit rate in the theorem and verify it numerically for the strongest censoring levels used in the simulations. revision: yes
Referee: The practical choice of the tuning parameter (>1) is left without guidance on how it should scale with sample size or censoring intensity to keep the linearization remainder controlled; the simulation section reports improved performance but does not include a sensitivity analysis or data-driven selection rule that would confirm robustness across the full censoring range.

Authors: We accept that additional guidance is warranted. In the revision we will add a subsection on tuning-parameter selection, recommending a scaling of the form 1 + c/log n with c chosen from a preliminary censoring-intensity estimate, together with a bootstrap-based data-driven rule that minimizes the estimated asymptotic variance. We will also expand the simulation study with a sensitivity table across a grid of tuning values and censoring intensities. revision: yes

Circularity Check

0 steps flagged

No circularity: asymptotics derived from regular variation via new process and linearization

full rationale

The paper constructs a weighted truncated Nelson-Aalen tail process with tuning parameter >1 and derives uniform Gaussian approximation plus asymptotic normality of the resulting integral estimators directly from standard regular-variation assumptions on the tail and censoring mechanism. No equation reduces the claimed normality or consistency to a fitted parameter, self-defined quantity, or prior self-citation chain; the linearization is an analytic step applied to the new functional rather than a renaming or tautology. The derivation remains self-contained against the stated external assumptions.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The claim rests on regular-variation assumptions for the tail, the existence of a suitable weighting function that restores the Gaussian limit under strong censoring, and a linearization identity whose validity is asserted but not derived in the abstract; the tuning parameter greater than one is a free choice whose effect on the approximation is not quantified.

free parameters (1)

tuning parameter
Indexing parameter larger than one that defines the family of estimators; its specific value is not derived from data or from a uniqueness theorem.

axioms (1)

domain assumption Tail distribution is regularly varying
Invoked to obtain the uniform Gaussian approximation and asymptotic normality.

invented entities (1)

weighted and truncated Nelson-Aalen tail empirical process no independent evidence
purpose: Restores tractable asymptotic structure over the entire censoring range
New object constructed in the paper to bypass the classical restriction to weak censoring.

pith-pipeline@v0.9.0 · 5561 in / 1320 out tokens · 44882 ms · 2026-05-14T17:45:21.318622+00:00 · methodology

discussion (0)

Reference graph

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