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REVIEW 3 major objections 4 minor 17 references

Three incomplete Tsallis-type entropy measures induce tests of stochastic equality against the increasing convex order, with asymptotic normality and competitive power against the earlier CRE test.

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2026-07-13 03:09 UTC pith:EXSSFOKU

load-bearing objection Solid incremental toolkit paper: three new incomplete-entropy orderings that correctly sit between st and icx, plus usable two-sample tests that often beat Zardasht 2015 in power. the 3 major comments →

arxiv 2607.09418 v1 pith:EXSSFOKU submitted 2026-07-10 math.ST stat.APstat.TH

Tests for Increasing Convex Ordering Based on Generalized Tsallis Entropy Measures

Aritra Saha , Siddhartha Chakraborty , Md. Zafar Anis This is my paper
classification math.ST stat.APstat.TH MSC 94A1760E1562G10
keywords Mean Residual LifeIncomplete Cumulative Residual EntropyIncreasing Convex OrderingAsymptotic NormalityTsallis entropynonparametric testsstochastic orders
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper shows that three incomplete entropy functionals—the weighted cumulative residual entropy and both the ordinary and weighted cumulative residual Tsallis entropies—preserve the increasing convex order. Because the complete-entropy difference is therefore nonnegative under the alternative and zero under equality, the corresponding sample estimators yield consistent two-sample tests. The tests are asymptotically normal under the null, their finite-sample size and power are examined by Monte Carlo on Weibull, gamma and Student-t families with equal means, and they are shown to be competitive with, and often more powerful than, the earlier incomplete CRE test. The work supplies a practical nonparametric toolkit for comparing lifetime or risk distributions when only second-order stochastic dominance is of interest.

Core claim

If X is smaller than Y in the increasing convex order, then the incomplete weighted CRE, incomplete CRTE and incomplete weighted CRTE of X are pointwise smaller than those of Y; the resulting complete-entropy differences therefore serve as discrepancy measures that vanish under stochastic equality and are strictly positive under the ordered alternative, and their studentized L-statistic estimators are asymptotically standard normal under the null.

What carries the argument

The three incomplete entropy functions expressed as expectations of convex (respectively increasing) kernels of the form ψ(t,X)I(X>t); these kernels convert the increasing-convex characterization into the entropy inequalities that underwrite the tests.

Load-bearing premise

The asymptotic variance formulas and the Monte-Carlo critical values remain valid for the continuous distributions used, including the heavy-tailed Pareto and Student-t examples, without explicit verification of the required second-moment conditions.

What would settle it

A continuous pair of distributions for which X ≤_icx Y yet the complete weighted CRE (or CRTE) difference is non-positive, or a Monte-Carlo experiment in which the studentized statistic fails to approach N(0,1) under equality for large equal sample sizes.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper defines incomplete weighted CRE (IWCRE), incomplete CRTE (ICRTE) and incomplete weighted CRTE (IWCRTE), introduces the associated partial orders, and proves that the usual stochastic order and the increasing convex order both imply each of these new orders (Theorems 3.2/3.4, 4.2/4.4, 5.2/5.4), with counter-examples showing the converses fail. Using the complete versions of the measures it constructs discrepancy statistics Δ_w, Δ_α and Δ_w_α for the two-sample problem H0: X =_st Y versus H1: X ≤_icx Y (X ≠_st Y), obtains asymptotic normality of the studentized estimators under the null by appealing to existing L-statistic theory, and evaluates finite-sample size and power by Monte Carlo (N_r = 10 000) under equal-mean Weibull, Gamma and Student-t alternatives, comparing the procedures with Zardasht’s CRE test.

Significance. If the technical claims hold, the work supplies a flexible family of entropy-based nonparametric tests for increasing-convex alternatives that can improve on the existing CRE test of Zardasht (2015) for several common lifetime models; the equal-mean simulation design is carefully chosen to isolate ageing/structural differences. The implication theorems rest on standard characterizations of st and icx orders applied to explicitly constructed increasing/convex kernels, and the asymptotic variance formulae are given in closed form. These are genuine, usable contributions to the nonparametric ordering literature, provided the remaining regularity and implementation gaps are closed.

major comments (3)
  1. [§5.1, Eq. (41)] Eq. (41) (§5.1): the estimator ˆσ²_X(J_{3α}) is written with the kernel J_2 and the double-sum structure of the WCRE variance (Eq. 25) rather than with J_{3α} and the factor 8 that appears in the population formula immediately above it. Consequently the studentized WCRTE statistic is not correctly studentized as claimed, rendering the asymptotic N(0,1) statement and all WCRTE critical values invalid until the formula is corrected.
  2. [Table 2] Table 2, n = m = 100, θ = 1 (null) row: the WCRTE empirical rejection rate is reported as 0.460 while the other three procedures stay near the nominal 0.05. This single entry is incompatible with the claim of proper Type-I control and, unless it is a typographical error, indicates a systematic implementation failure that contaminates every WCRTE power figure in the table and undermines the comparative conclusions drawn for the weighted Tsallis procedure.
  3. [§5 and §6] §5 (representation and Definition 5.1) and the Student-t / Pareto examples of §6: the incomplete IWCRTE is introduced with ξ^w_α but Definition 5.1 switches to T^w_α; the subsequent “proof” of the expectation representation (Eqs. 36–37) is written for the complete measure and uses incorrect integral limits. In addition the L-statistic asymptotics invoked for the weighted procedures require finite second moments of kernels that involve factors of x or x²; these fail for the Student-t distribution with 2 degrees of freedom (and for the Pareto counter-examples with small shape) that are used both for counter-examples and for power evaluation. The paper never states or verifies the needed moment conditions, so the asymptotic justification does not cover the very distributions appearing in the numerical study.
minor comments (4)
  1. [§6] Repeated sentence in the opening paragraph of §6 (“The empirical power of the procedures was evaluated using N_r = 10 000 imes 2”).
  2. [§§3–5] Inconsistent notation: incomplete measures are denoted ξ^w, T_α, ξ^w_α while Definition 5.1 and later displays switch to T^w_α; the estimator in §3.1 is written ˆξ(X) rather than ˆξ^w(X).
  3. [throughout] Typographical errors: “discripency” (p. 6), “tsallis” left uncapitalized in Definition 4.1, MSC code “62G1” incomplete, and the arXiv date “July 13, 2026” is in the future.
  4. [§6] The precise method used to obtain the “simulated critical threshold” (pooled null, separate null samples, or asymptotic quantiles) is never stated; a one-sentence clarification would make the power study reproducible.

Circularity Check

1 steps flagged

No significant circularity: orderings follow from standard stochastic-order characterizations applied to explicitly constructed kernels; tests are ordinary L-statistic discrepancy measures whose asymptotics rest on independent prior estimators.

specific steps
  1. self citation load bearing [Section 3.1 (estimator of ξ_w) and Section 5 (estimator of ξ_w_α)]
    "Chakraborty and Nanda (2025) proposed an estimator for ξ_w(X) as ξ̂(X)=-½n ∑ X^{2}_{(i)} J_{2}(i/(n+1)) … As given in Chakraborty and Nanda (2025), we can express ξ_w_α(X) as … The plug-in estimator …"

    The asymptotic normality of the proposed test statistics is taken from the authors’ own concurrent arXiv note on estimation of weighted cumulative residual Tsallis entropy. The citation is not load-bearing for the ordering theorems (which rest on classical stochastic-order characterizations), but it is the sole source of the variance formulae used for studentization; hence a minor self-citation that does not force the central claim.

full rationale

The paper defines incomplete weighted CRE, incomplete CRTE and incomplete WCRTE (Eqs. 15, 26, 35), rewrites them as expectations of kernels ψ_w, ψ_α, ψ_w_α, then verifies that those kernels are increasing (hence st-order preserves the incomplete measures) and convex (hence icx-order preserves them). The three implication theorems (3.2/3.4, 4.2/4.4, 5.2/5.4) are therefore ordinary applications of the classical characterizations of ≤_st and ≤_icx; nothing is defined in terms of a later “prediction.” The discrepancy measures Δ_w, Δ_α, Δ_w_α are simply the complete-entropy differences, which vanish under equality and are non-negative under the alternative by the preceding implications. Their estimators are plug-in L-statistics taken from earlier independent work (Chakraborty & Nanda 2025, Zardasht 2019); the asymptotic normality statements are the usual ones for such L-statistics once second-moment conditions hold. Self-citations appear only for the estimators and for the baseline CRE test of Zardasht (2015); they are not load-bearing for the ordering theorems themselves. No fitted parameter is later re-labeled a prediction, no uniqueness theorem is imported from the authors’ prior work, and no known empirical pattern is merely renamed. The single minor self-citation of the authors’ own estimator paper does not raise the score above 1.

Axiom & Free-Parameter Ledger

2 free parameters · 3 axioms · 0 invented entities

The paper rests on standard stochastic-order characterizations, continuous non-negative random variables, and the asymptotic theory of L-statistics already established for the complete entropy functionals. The only free parameters are the Tsallis order α (chosen by power maximization) and the usual sample-size ratio ρ. No new physical or mathematical entities are postulated.

free parameters (2)
  • Tsallis order α = 1.3 / 1.5
    Selected after simulation to maximize empirical power (α=1.3 for CRTE, α=1.5 for WCRTE); not derived from first principles.
  • sample-size ratio ρ = n/N = 0.5
    Appears in the asymptotic variance formula; fixed at 1/2 by the equal-sample-size design.
axioms (3)
  • standard math X ≤_st Y iff E[h(X)] ≤ E[h(Y)] for all increasing h with existing expectations
    Invoked in Theorems 3.2, 4.2, 5.2 to transfer the usual stochastic order to the new entropy orders.
  • standard math X ≤_icx Y iff E[h(X)] ≤ E[h(Y)] for all convex h with existing expectations
    Invoked in Theorems 3.4, 4.4, 5.4; the second-derivative calculations show the relevant integrands are convex.
  • domain assumption Underlying distributions are continuous and non-negative with finite second moments needed for L-statistic asymptotics
    Required for the asymptotic normality statements and for the variance estimators (25),(34),(41) to be consistent; never verified for the heavy-tailed examples.

pith-pipeline@v1.1.0-grok45 · 19131 in / 2648 out tokens · 38472 ms · 2026-07-13T03:09:08.256726+00:00 · methodology

0 comments
read the original abstract

In this paper, we study several incomplete entropy measures, namely the Incomplete Weighted Cumulative Residual Entropy, the Incomplete Cumulative Residual Tsallis Entropy and its weighted version, and introduce the associated partial orderings. Their connections with certain well-known stochastic orderings are also investigated. Based on these characterizations, a class of nonparametric tests for stochastic equality against ordered alternatives is developed. The asymptotic properties of the proposed tests are derived, while their finite-sample performances are assessed through extensive Monte Carlo simulations under various alternative models and sample sizes. These tests are further compared with the test based on incomplete cumulative residual entropy proposed by Zardasht (2015).

Figures

Figures reproduced from arXiv: 2607.09418 by Aritra Saha, Md. Zafar Anis, Siddhartha Chakraborty.

Figure 1
Figure 1. Figure 1: Graphical comparison of the considered measures of proposed entropies for the [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗

discussion (0)

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Reference graph

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