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arxiv: 2604.15061 · v1 · submitted 2026-04-16 · 🧮 math.ST · stat.TH

On general weighted cumulative residual (past) extropy of extreme order statistics

Santosh Kumar Chaudhary , Sarikul Islam , Nitin Gupta This is my paper

Pith reviewed 2026-05-10 09:21 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords weighted cumulative residual extropyextreme order statisticscharacterization theoremsgeneralized Pareto distributionpower distributionreliability modelssurvival analysisdynamic extropy
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The pith

General weighted cumulative residual and past extropy measures for the smallest and largest order statistics uniquely characterize the underlying distribution.

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

This paper defines the general weighted cumulative residual extropy for the smallest order statistic and the general weighted cumulative past extropy for the largest order statistic, along with dynamic versions of both. It proves that these measures and their dynamic forms uniquely determine the parent distribution when the variables are absolutely continuous. The authors also derive explicit characterization results that identify the generalized Pareto distribution and the power distribution from these quantities. A reader would care because the results supply new information-theoretic tools for distinguishing reliability models from data on extreme lifetimes in survival analysis and engineering.

Core claim

The authors introduce the general weighted cumulative residual extropy (GWCREx) of the smallest order statistic and the general weighted cumulative past extropy (GWCPEx) of the largest order statistic, together with their dynamic counterparts. They prove that these weighted measures and the dynamic versions uniquely characterize the underlying distribution. They further obtain new characterization results that identify the generalized Pareto distribution and the power distribution via these same quantities.

What carries the argument

The general weighted cumulative residual extropy (GWCREx) for the minimum order statistic and the general weighted cumulative past extropy (GWCPEx) for the maximum order statistic, which extend weighted cumulative residual and past entropy concepts to extropy and function as unique identifiers of the parent distribution from its extremes.

Load-bearing premise

The random variables must be absolutely continuous with positive density on their support and the weighting function must be positive and integrable.

What would settle it

Two distinct absolutely continuous distributions with positive densities that produce identical values of the general weighted cumulative residual extropy for their smallest order statistics would falsify the uniqueness claim.

read the original abstract

Weighted extropy has recently emerged as a flexible information measure for quantifying uncertainty, with particular relevance to order statistics. In this paper, we introduce and study a weighted cumulative analogue of extropy, extending the framework of weighted cumulative residual and cumulative past entropies to extreme order statistics. Specifically, we define the general weighted cumulative residual extropy (GWCREx) for the smallest order statistic and the general weighted cumulative past extropy (GWCPEx) for the largest order statistic, along with their dynamic versions. We show that these weighted measures and their dynamic counterparts uniquely characterize the underlying distribution. Moreover, we establish new characterization results for two widely used reliability models: the generalized Pareto distribution and the power distribution. The proposed framework provides a unified information-theoretic tool for analysing extreme lifetimes in reliability engineering and survival analysis.

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 / 3 minor

Summary. The paper defines the general weighted cumulative residual extropy (GWCREx) for the smallest order statistic and the general weighted cumulative past extropy (GWCPEx) for the largest order statistic, along with their dynamic versions. It proves that these functionals (and their dynamic counterparts) uniquely characterize the underlying absolutely continuous distribution under the stated regularity conditions on the weight function. As corollaries, it derives explicit characterization results for the generalized Pareto distribution and the power distribution.

Significance. If the uniqueness arguments hold, the work supplies new information-theoretic tools for characterizing distributions via weighted cumulative extropy measures applied to extreme order statistics. This is relevant to reliability engineering and survival analysis, where extreme lifetimes are of interest, and the specific results for the GPD and power families provide concrete, falsifiable characterizations that can be checked against data.

major comments (2)
  1. [Theorems on uniqueness and dynamic versions] The uniqueness proofs (presumably in the main theorems following the definitions of GWCREx and GWCPEx) proceed by equating the functionals for two distributions and recovering the survival function via an integral equation. The manuscript should explicitly state and verify the conditions under which this inversion is unique (e.g., by exhibiting the recovery formula or showing that the weight function yields an invertible kernel), particularly for the dynamic versions where the conditioning on time introduces additional boundary terms.
  2. [Characterization results for GPD and power distribution] For the characterization corollaries of the generalized Pareto and power distributions, the paper substitutes the known survival functions and verifies the functional equation. It should also confirm that no other distributions in the assumed class satisfy the same equation, to ensure the characterizations are sharp rather than merely necessary conditions.
minor comments (3)
  1. [Introduction and definitions] Notation for the general weight function w(·) and its integrability condition should be introduced once at the beginning and used consistently; currently it appears to be redefined in each section.
  2. [Introduction] The abstract claims 'new characterization results' for GPD and power; the introduction should briefly contrast these with existing characterizations based on cumulative residual entropy or extropy to clarify the incremental contribution.
  3. [Numerical examples or illustrations] Figure captions (if any) and table headings for numerical illustrations of the measures should include the specific weight functions used, to allow reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and the helpful comments. We address each major comment below and will revise the paper accordingly to improve clarity.

read point-by-point responses

  1. Referee: [Theorems on uniqueness and dynamic versions] The uniqueness proofs (presumably in the main theorems following the definitions of GWCREx and GWCPEx) proceed by equating the functionals for two distributions and recovering the survival function via an integral equation. The manuscript should explicitly state and verify the conditions under which this inversion is unique (e.g., by exhibiting the recovery formula or showing that the weight function yields an invertible kernel), particularly for the dynamic versions where the conditioning on time introduces additional boundary terms.

    Authors: We agree that the invertibility conditions merit explicit statement. In the proofs, equating the measures for two distributions yields an integral equation whose kernel is the weight function times the survival function (or its conditional version). Under the maintained regularity conditions (weight function positive, continuous, and integrable as stated in the paper), differentiation recovers the survival function uniquely. For the dynamic versions, the lower integration limit is the conditioning time t, so boundary terms at t vanish identically by construction of the conditional survival function. In the revised manuscript we will add a short remark after each main theorem that displays the explicit recovery formula and confirms uniqueness of the inversion. revision: yes

  2. Referee: [Characterization results for GPD and power distribution] For the characterization corollaries of the generalized Pareto and power distributions, the paper substitutes the known survival functions and verifies the functional equation. It should also confirm that no other distributions in the assumed class satisfy the same equation, to ensure the characterizations are sharp rather than merely necessary conditions.

    Authors: The corollaries are direct consequences of the uniqueness theorems already proved for the general weighted measures. Because those theorems establish that the functional determines the distribution uniquely within the class of absolutely continuous distributions satisfying the regularity conditions, the characterizing equation holds if and only if the distribution belongs to the stated family. To make this explicit we will append one clarifying sentence to each corollary noting that sharpness follows from the preceding uniqueness result. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained functional inversions

full rationale

The central results express the new GWCREx and GWCPEx functionals (and dynamic versions) explicitly in terms of the survival function or CDF, then recover the parent distribution by solving the resulting integral or differential equation under the stated regularity conditions (absolute continuity, positive density, positive integrable weight). These inversion steps are standard and independent of the definitions themselves. Characterizations of the generalized Pareto and power distributions arise as direct corollaries by substitution of the known survival functions. No load-bearing self-citation, fitted-parameter prediction, ansatz smuggling, or self-definitional reduction appears; the uniqueness arguments rest on external mathematical properties of the functional equations rather than on the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The paper introduces two new functionals (GWCREx and GWCPEx) as definitions rather than deriving them from prior quantities. No numerical parameters are fitted. The work relies on standard assumptions of continuous distributions and positive weighting functions.

axioms (2)
  • domain assumption The underlying random variables are i.i.d. absolutely continuous with positive density on an interval support.
    Required for the order statistics to have well-defined densities and for the integral expressions of the weighted extropy measures to be finite.
  • domain assumption The weighting function w(x) is positive and integrable over the support.
    Invoked to ensure the weighted cumulative measures are well-defined and to obtain the uniqueness results.
invented entities (2)
  • General weighted cumulative residual extropy (GWCREx) for the smallest order statistic no independent evidence
    purpose: Quantify weighted uncertainty in residual lifetime of the minimum order statistic
    Newly defined functional; no independent empirical evidence provided beyond the mathematical construction.
  • General weighted cumulative past extropy (GWCPEx) for the largest order statistic no independent evidence
    purpose: Quantify weighted uncertainty in past lifetime of the maximum order statistic
    Newly defined functional; no independent empirical evidence provided beyond the mathematical construction.

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