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Extropy of consecutive-k-out-of-n systems derived and shown to characterize distributions

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2026-07-09 22:01 UTC pith:TTIUO3DA

load-bearing objection Solid but incremental reliability-theory paper with correct math and sloppy presentation the 4 major comments →

arxiv 2607.06992 v1 pith:TTIUO3DA submitted 2026-07-08 math.ST stat.TH

A Study on Cumulative Residual Extropy of Linear Consecutive k-out-of-n:G Systems

Aman Pandey , Chanchal Kundu This is my paper
classification math.ST stat.TH
keywords residualcumulativeextropyproposedconsecutivedynamicestimatork-out-of-n
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 derives an explicit integral representation for the cumulative residual extropy (CREx) of a linear consecutive k-out-of-n:G system — a reliability structure that functions when at least k consecutive components are working. Under the condition 2k ≥ n, which simplifies the system's survival function to a linear combination of component survival probabilities, the authors express the system's information-theoretic uncertainty as an integral involving the squared system survival function weighted by the quantile density of the component lifetime distribution. They then prove that this measure serves as a characterization tool: equality of CREx between two such systems built from different component distributions holds if and only if those distributions differ by a location shift (Theorem 2.5), and a ratio-based equality characterizes the broader location-scale family (Theorem 2.6). The paper also establishes stochastic ordering results — showing, for instance, that under a decreasing failure rate condition, the hazard rate order between systems is reversed by CREx — derives upper and lower bounds, introduces a dynamic (residual-life) version of the measure, and proposes a nonparametric estimator with proven strong consistency.

Core claim

The central result is the explicit representation J(T_{k|n:G}) = -1/2 ∫_0^1 Φ(Ḡ_{k|n:G}(v)) f(F^{-1}(v)) dv, where Ḡ_{k|n:G}(v) = (n-k+1)(1-v)^k - (n-k)(1-v)^{k+1} is the survival function of the system lifetime expressed through the probability integral transform, and f(F^{-1}(v)) is the quantile density of the component distribution. This representation reduces the system-level uncertainty measure to a one-dimensional integral over (0,1) involving only the component quantile density and a polynomial function of v determined by k and n. The characterization theorems then show that this integral is injective with respect to location shifts and location-scale transformations of the component,

What carries the argument

The probability integral transformation v = F(w) converts the system survival function into a polynomial Ḡ_{k|n:G}(v) on (0,1) when 2k ≥ n; the Vasicek-type finite-difference approximation of the quantile derivative dF^{-1}(v)/dv using spacings of order statistics enables nonparametric estimation; the dispersive order X ≤_disp Y and the location-independent riskier order provide the stochastic comparison framework; the DFR (decreasing failure rate) property connects hazard rate ordering to CREx ordering.

Load-bearing premise

All results depend on the condition 2k ≥ n, which ensures that two non-overlapping runs of k functioning components cannot coexist, thereby reducing the system survival function to a simple linear combination of powers of the component survival function. This restricts the applicability to roughly the upper half of the (k, n) parameter space — systems where k is at least half of n.

What would settle it

If one attempts to apply the representation in Equation 2.3 to a system with 2k < n (say k=2, n=10), the survival function Ḡ_{k|n:G}(v) given in (2.1) no longer equals the true system survival function, so the integral would not yield the correct CREx value. The characterization theorems would also fail because the injectivity argument depends on the specific polynomial form of Ḡ_{k|n:G}, which is only valid under 2k ≥ n.

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

If this is right

  • The explicit integral representation allows closed-form or semi-analytical computation of system uncertainty for any component distribution whose quantile function is known, bypassing the need for system-level simulation.
  • The characterization results imply that CREx can distinguish between location and location-scale families of component distributions through system-level observations alone, which could inform component selection in reliability design.
  • The nonparametric estimator extends to any coherent system whose survival function can be expressed as a function of the component survival function, suggesting applicability beyond consecutive-k-out-of-n structures.
  • The dynamic CREx version provides a tool for monitoring how system uncertainty evolves as the system ages, which is relevant for condition-based maintenance scheduling.

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

4 major / 8 minor

Summary. This paper studies the cumulative residual extropy (CREx) of linear consecutive k-out-of-n:G (C(k|n:G)) systems under the condition 2k >= n. The authors derive an explicit integral representation for the CREx of the system lifetime (Eq. 2.3), establish bounds and characterization results under stochastic orders (Theorems 2.1-2.6), introduce a dynamic version (Section 3), and propose a nonparametric estimator with consistency proof (Section 4). The core derivation is a straightforward application of the probability integral transform to the known survival function of C(k|n:G) systems (Eq. 1.3, from Eryilmaz 2009). The characterization results (Theorems 2.5, 2.6) connect equality of CREx to location and location-scale families via the dispersive order. Simulation studies and a real-data illustration are provided.

Significance. The paper applies the relatively new CREx measure to an important class of reliability systems. The explicit representation (Eq. 2.3) and the characterization results (Theorems 2.5-2.6) are the main theoretical contributions. The nonparametric estimator follows the Vasicek (1976) spacing approach and its consistency proof is standard but correct. The results are a natural extension of recent work by Kayid and Alshehri (2024) and Kayid and Balakrishnan (2025) on entropy and cumulative residual entropy for the same systems. The contribution is incremental but solid for a specialist audience.

major comments (4)
  1. Theorem 3.2 (Section 3): The statement claims bounds M_1 * J(s) <= J(T^s_{k|n:G}) <= M_2 * J(s), where M_1 = inf Phi(R(u))/Phi(u) and M_2 = sup Phi(R(u))/Phi(u). However, since Phi(t) = -1/2 t^2 is a decreasing function of t (for t > 0), if R(u) >= u then Phi(R(u)) <= Phi(u), so the ratio Phi(R(u))/Phi(u) <= 1. The issue is that the proof defines a signed measure d_mu_s(u) involving Phi(u)/f(F^{-1}(1-u*F_bar(s))), but Phi(u) is negative. This means d mu_s is a signed (or negative) measure, and the standard argument that M_1 <= ratio <= M_2 implies M_1 * integral <= integral of ratio <= M_2 * integral requires careful handling of signs. The authors should verify that the sign conventions are consistent throughout the proof and clarify this point.
  2. Example 2.1 (Section 2): The text states the component lifetimes follow a 'Lomax distribution' but the CDF given is F(w) = 1 - exp(1 - (w/eta)^beta), which is actually a reversed Weibull (or type III extreme value) distribution, not a Lomax. The Lomax CDF is F(w) = 1 - (1 + w/eta)^{-beta}. The figure caption also says 'Weibull distribution.' This inconsistency between the stated distribution name and the formula needs to be corrected. Example 2.2 correctly uses the Lomax CDF, so Example 2.1 appears to have a copy-paste or labeling error.
  3. Introduction, final paragraph: The paper outline mentions 'Section 4: an application of the proposed CREx measure in image processing' and 'Section 5: dynamic version of CREx.' However, the actual manuscript has Section 3 as 'Dynamic CREx,' Section 4 as 'Nonparametric estimation,' and Section 5 as 'Conclusion.' There is no image processing application section, and the section numbering in the outline does not match the actual structure. This should be corrected.
  4. Theorem 2.2 (Section 2): The condition that Phi(t) := Phi(G_bar_{k|n:G}(s))/s is decreasing in s is stated as an assumption. The authors should verify whether this condition is automatically satisfied for the specific form of G_bar_{k|n:G} given in Eq. (2.1), or at minimum provide a remark on when it holds. If it is not automatically satisfied, the practical applicability of Theorem 2.2 is limited without further analysis of this condition for specific system parameters (k, n).
minor comments (8)
  1. Example 2.2: The Lomax CDF is written with parameters 'beta' and 'eta' in the text but the sentence below says 'where beta and eta represent the shape and scale parameters, respectively,' while the CDF formula line itself says 'w > 0, beta > 0, eta > 0' which is redundant. The parameter naming should be consistent with the formula.
  2. Section 4.1: The simulation study uses Lomax distribution with parameters (alpha, lambda) in the text description but (beta, eta) in the CDF formula. The notation should be unified.
  3. Abstract: The phrase 'Fromaninferentialperspective' appears to be a formatting artifact (missing spaces). Should read 'From an inferential perspective.'
  4. Section 3 title: 'Dynamic CREx for C(k|n:G) system' — the section numbering in the introduction's outline (Section 5 for dynamic CREx) does not match the actual structure (Section 3).
  5. Equation (2.4) in the proof of Theorem 2.2: The text references 'relation (2.4)' but this appears to refer to the chain of implications involving dispersive and lir orders, which is labeled (2.4) earlier. The cross-referencing is correct but could be clearer.
  6. Theorem 2.3: The roles of b_1 and b_2 as lower and upper bounds should be verified. Since Phi is negative and decreasing, the sup and inf of the ratio Phi(G_bar)/Phi(1-v) need careful sign handling. A brief remark clarifying the direction of the bounds would help readers.
  7. Table 5: The absolute errors for (n,k) = (5,3), (5,4), (5,5) are notably larger than for other configurations. The authors acknowledge this but the explanation (tail observations) could be more precise.
  8. Reference formatting: Several references have minor formatting issues (e.g., 'Eryilmaz, S. (2010). Conditional lifetimes of consecutive k-out-of-nSystems, IEEE Transactions on Reliability. 59(1)' has missing spaces).

Circularity Check

0 steps flagged

No circularity: derivation chain is self-contained with externally-sourced building blocks

full rationale

The paper's derivation chain is entirely non-circular. The starting ingredients are: (1) the CREx definition (Eq. 1.2, from Jahanshahi et al. 2019 — external), (2) the survival function of C(k|n:G) systems under 2k≥n (Eq. 1.3, from Eryilmaz 2009 — external), and (3) standard stochastic order characterizations (from Shaked & Shanthikumar 2007 — external). The main representation (Eq. 2.3) is obtained by substituting (1.3) into (1.2) and applying the probability integral transformation — a direct mathematical operation, not a fit or a definitional identity. The characterization theorems (2.5, 2.6) derive their conclusions from the dispersive order property f_X(F_X^{-1}(v)) ≥ f_Y(F_Y^{-1}(v)) and integral sign arguments, not from circularly assuming the conclusion. The nonparametric estimator (Eq. 4.1) is constructed from the theoretical representation using the Vasicek (1976) spacing method — an external methodology — and its consistency is proved via the Strong Law of Large Numbers, not by calibrating to match the target. No self-citations appear among the load-bearing references (Pandey and Kundu do not cite their own prior work for any foundational result). The skeptic's concern about a possible sign error in Theorem 2.5 is a correctness question, not a circularity issue. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The paper introduces no new entities, particles, or postulated objects. It applies existing mathematical definitions (CREx) to existing system models (C(k|n:G)) using standard stochastic order tools. The free parameters are simulation/estimation tuning choices, not theoretical constructs.

free parameters (2)
  • Window parameter m = floor(sqrt(N))
    Chosen as m = floor(sqrt(N)) in the simulation study for the spacing-based estimator. This is a tuning parameter, not fitted to data, but it is a free choice affecting estimator performance.
  • Lomax distribution parameters (beta, eta) = Various: (2.5, 1.0), (3.5, 2.5), (1.5, 4.5)
    Used as simulation settings, not fitted. In the real data analysis, beta=1.3938, eta=5.5852 are MLE estimates from the COVID-19 data.
axioms (5)
  • domain assumption Component lifetimes X_1, ..., X_n are iid with common survival function S(w).
    Stated in the introduction and used throughout. This is a standard assumption in reliability theory but limits applicability to homogeneous, independent components.
  • domain assumption The condition 2k >= n holds.
    Invoked in Section 2 to use the simplified survival function (Eq. 1.3) from Eryilmaz (2009). All derived expressions, bounds, and characterizations depend on this condition.
  • standard math The survival function of C(k|n:G) for 2k >= n is given by Eq. (1.3).
    Cited from Eryilmaz (2009). This is the foundational result from which all CREx expressions are derived.
  • standard math CREx is defined as J(X) = -1/2 * integral of F_bar^2(w) dw (Eq. 1.2).
    Cited from Jahanshahi et al. (2019). This is the definition of the measure being applied.
  • standard math X <= disp Y implies f_Y(F_Y^{-1}(u)) <= f_X(F_X^{-1}(u)) for 0 < u < 1.
    Standard characterization of the dispersive order, used in Theorems 2.5 and 2.6. Cited from Shaked and Shanthikumar (2007).

pith-pipeline@v1.1.0-glm · 23120 in / 2675 out tokens · 224109 ms · 2026-07-09T22:01:42.064620+00:00 · methodology

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read the original abstract

In this article, we investigate the cumulative residual extropy associated with linear consecutive k-out-of-n:G systems, which play an important role in reliability theory and engineering applications. We first derive explicit expressions for the proposed measure and examine the behavior of cumulative residual extropy under a variety of stochastic orderings. In addition, several bounds and meaningful results of the characterization are established. Moreover, we also introduced the dynamic version of the cumulative residual extropy and explored the relationship between the proposed dynamic version and the mean residual life function. Fromaninferentialperspective, wedevelopanonparametricestimationprocedure for the cumulative residual extropy and establish the corresponding consistency properties of the estimator. The finite-sample performance of the proposed estimator is further investigated through extensive Monte Carlo simulation studies under different parametric settings and validated through a real dataset.

Figures

Figures reproduced from arXiv: 2607.06992 by Aman Pandey, Chanchal Kundu.

Figure 1
Figure 1. Figure 1: The plots of CREx for C(k|n:G) systems under the Weibull distribution. Now, we recall some stochastic orders, which will be used to obtain bounds and characteri￾zations. For more details, one may refer to Shaked and Shanthikumar (2007). Let X and Y be RVs with PDFs fX, fX and SFs F¯X, F¯ Y , respectively. 1. Stochastic Order. X1 is said to be smaller than X2 in the usual stochastic order, denoted by X ≤st … view at source ↗
Figure 2
Figure 2. Figure 2: Graphical illustration of b1 · J(X), J(Tk|n:G), and b2 · J(X) for k = 5, 6, ..., 10. The following result establishes two new bounds for the CREx measure of consecutive sys￾tems. Theorem 2.4. Assume that X is an absolutely continuous RV with support set A, and denote M1 = infw∈A f(w) and M2 = supw∈A f(w), where f is PDF of X. Then, the CREx satisfies J(Vk|n:G) M2 ≤ J(Tk|n:G) ≤ J(Vk|n:G) M1 , where J(Vk|n:G… view at source ↗
Figure 3
Figure 3. Figure 3: The plot of DCREx for a C(k|10:G) system when component lifetimes follow the exponential distribution with parameter γ ∈ (0, 3), for k = 5, . . . , 10. It is evident from [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

discussion (0)

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Reference graph

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