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REVIEW 4 major objections 5 minor 25 references

Ordering Results between Two Extreme Order Statistics with Heterogeneous Linear Failure Rate Distributed Components

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For independent linear-failure-rate components, comparing each component's two parameters componentwise is sufficient to order the lifetimes of the assembled series and parallel systems.

desk verdict True stochastic-order results for LFR components, but the Section 4 proofs mishandle the derivatives and need a careful rewrite before the paper is publishable. read the letter →

arxiv 2506.05773 v1 pith:QLRLBM66 submitted 2025-06-06 math.ST stat.TH

CM Revathi , Rajesh Moharana , Raju Bhakta This is my paper
classification math.STstat.TH MSC 60E1562G3090B25
keywords LinearFailureRateDistributionSeriesSystemParallelStochasticOrdersMagnitudeTransformVariabilityHeterogeneousComponents
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes sufficient conditions, read directly off the two parameters of each component, under which one series or parallel system is stochastically better than another when all component lifetimes are independent and follow linear failure rate (LFR) distributions. If every component in system X has both LFR parameters at least as large as the corresponding component in system Y, the paper proves X's series lifetime is smaller than Y's in the usual stochastic and hazard rate orders, and also in the likelihood ratio order when the scale parameter is common. A parallel-system analogue gives the usual stochastic order under the same componentwise parameter dominance. For series systems with a common scale parameter, the paper further derives the dispersive order from one shape-parameter inequality and the star, Lorenz, and convex-transform orders from the reverse inequality. Because the conditions are componentwise, a reliability engineer can rank two candidate designs by inspecting parameter tables rather than simulating system lifetimes.

What carries the argument

The load-bearing object is the linear failure rate distribution $\mathrm{LFR}(\alpha,\beta)$ with CDF $F(x)=1-e^{-(\alpha x+\beta x^2/2)}$, whose series-system survival and hazard are additive in the component parameters. Its quantile function, obtained by solving a quadratic, is $F^{-1}(p)=(-\sum\alpha_k+\sqrt{(\sum\alpha_k)^2-2\sum\beta_k\log(1-p)})/\sum\beta_k$, which lets transform-order conditions be checked by differentiation. The ordering implications $\le_{lr}\Rightarrow\le_{hr}\Rightarrow\le_{st}$ and $\le_c\Rightarrow\le_*\Rightarrow\le_{\mathrm{Lorenz}}$ then convert pointwise parameter inequalities into the desired system comparisons.

What would settle it

Take two two-component series systems with $\beta=1$, $\alpha=(2,4)$, and $\alpha^*=(1,3)$, which satisfy the parameter inequalities of Theorem 3.3, and plot $g_{Y_{1:2}}(x)/f_{X_{1:2}}(x)$ for $x>0$; if the ratio is not monotone increasing, the asserted likelihood-ratio ordering fails.

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Extended reading notes

Core claim

The paper's central claim is that the linear failure rate model makes system-level comparisons reducible to parameter-level comparisons. Writing each component as $X_k \sim \mathrm{LFR}(\alpha_k,\beta_k)$ with survival $\bar F_{X_k}(x)=\exp\{-\alpha_k x-\beta_k x^2/2\}$, the series system has survival $\exp\{-\sum_{k=1}^n(\alpha_k x+\beta_k x^2/2)\}$ and hazard $\sum_{k=1}^n(\alpha_k+\beta_k x)$. Therefore $\alpha_k\ge\alpha^*_k$ and $\beta_k\ge\beta^*_k$ for all $k$ imply $\bar F_{X_{1:n}}(x)\le \bar F_{Y_{1:n}}(x)$ and $h_{X_{1:n}}(x)\ge h_{Y_{1:n}}(x)$, giving the usual stochastic and hazard rate orders, and with a common $\beta$ the same shape dominance is shown to imply the likelihood ratio order. For parallel systems, the same componentwise dominance gives $X_{n:n}\le_{st}Y_{n:n}$. The transform-order theorems use the closed-form quantile function of the series system to convert parameter inequalities into monotonicity or convexity of $G^{-1}_{Y_{1:n}}(F_{X_{1:n}}(x))$.

Load-bearing premise

The arguments assume component lifetimes are independent, and the transform-order proofs additionally treat the minimum of independent LFR variables as itself LFR with parameters equal to the sums of the component parameters; the ordering results depend on those assumptions holding.

Editorial extensions

If this is right

  • A design team comparing two candidates only needs to check whether each component's $(\alpha,\beta)$ dominates the other's; if so, the entire series system is ordered without Monte Carlo simulation.
  • Because $X_{1:n}\le_{st}Y_{1:n}$ and $X_{1:n}\le_{hr}Y_{1:n}$ are established, the system with smaller component parameters is the one to choose when a longer series lifetime is the goal.
  • For parallel systems, componentwise dominance orders the maximum lifetime in the usual stochastic order, so redundancy decisions can be made from parameter tables.
  • The likelihood-ratio result under common $\beta$ gives a stronger ordering than the hazard or usual stochastic orders, useful for comparing failure distributions in full.
  • The transform-order results imply shape comparisons under a common scale, so variability and inequality of lifetime distributions are also ranked by the paper's conditions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper's examples, the proofs of the magnitude-order results are written for general $n$, so the parameter-table decision rule should apply to series and parallel systems of any size, not only to the $n=3$ cases plotted.
  • Because the series-system expressions depend only on the sums $\sum\alpha_k$ and $\sum\beta_k$, the same stochastic-order statements would survive if the heterogeneous components were permuted, suggesting an exchangeability property the paper does not state.
  • The counterexamples show that if one parameter dominates while the other reverses, the ordering can fail; this points toward a testable conjecture that full componentwise dominance is also necessary, not merely sufficient, for the hazard and likelihood-ratio orders.
  • Specializing $\beta_k=0$ recovers exponential-component comparisons and $\alpha_k=0$ recovers Rayleigh-component comparisons, so the theorems nest previously studied models as boundary cases.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper studies stochastic comparisons of series and parallel systems whose component lifetimes are independent but heterogeneous linear failure rate (LFR) random variables, with X_k ~ LFR(α_k, β_k) and Y_k ~ LFR(α*_k, β*_k). It claims: componentwise dominance α_k ≥ α*_k and β_k ≥ β*_k implies X_{1:n} ≤_st Y_{1:n} (Theorem 3.1), X_{1:n} ≤_hr Y_{1:n} (Theorem 3.2), and X_{n:n} ≤_st Y_{n:n} (Theorem 3.4); with a common β and α_k ≥ α*_k, the likelihood ratio order X_{1:n} ≤_lr Y_{1:n} holds (Theorem 3.3). Under the reverse shape ordering α*_k ≥ α_k with a common β, the paper claims the dispersive, star, and convex transform orders X_{1:n} ≤_disp Y_{1:n}, X_{1:n} ≤_* Y_{1:n}, and X_{1:n} ≤_c Y_{1:n} (Theorems 4.1–4.3). Section 5 provides numerical examples and counterexamples for each theorem, and Section 6 gives a brief application and conclusion.

Significance. If the theorems stand, the paper contributes simple, parameter-based sufficient conditions for comparing series and parallel systems with heterogeneous LFR components, including transform orders that are not otherwise available for this family. My own calculations confirm the theorem statements: the Section 3 results are elementary and correct, and the Section 4 theorems are true despite the defective proofs. The paper is self-contained, does not rely on fitted parameters or circular reasoning, and the examples and counterexamples are consistent with the claimed orders. The value of the contribution is real but is currently compromised by incorrect proof apparatus in Section 4 and one faulty derivative display in Section 3.

major comments (4)
  1. [Section 4, Theorem 4.1, Eq. (4.6)] The derivative displayed in Eq. (4.6) is not the derivative of G^{-1}_{Y_{1:n}}(p) - F^{-1}_{X_{1:n}}(p). Differentiating the quantile in Eq. (4.4) gives q'_A(p) = 1/[(1-p) sqrt(A^2 - 2 n β log(1-p))] with A = Σ α_k, so the correct derivative is (1/(1-p)) [ (A^{*2} - 2nβ log(1-p))^{-1/2} - (A^2 - 2nβ log(1-p))^{-1/2} ]. The printed expression has the square roots in the numerator rather than the denominator and omits the factor 1/(1-p); as written it is negative when A ≥ A*, so the sign conclusion drawn after Eq. (4.7) is reversed. The theorem is true, but the proof as printed does not establish it.
  2. [Section 4, Theorem 4.2 proof] The star order is defined by monotonicity of G^{-1}_{Y_{1:n}}(F_{X_{1:n}}(x))/x, so the proof must show that the derivative of this ratio is nonnegative. The displayed derivative in the proof is not the derivative of the ratio in Eq. (4.8); differentiating that ratio produces terms involving x Q'(x) - Q(x), not the expression shown. A derivative of G^{-1}_{Y_{1:n}}(F_{X_{1:n}}(x)) alone, even if positive, does not imply the star order. The theorem statement may be true, but the proof as written is incomplete and the displayed derivative is not justified.
  3. [Section 4, Theorem 4.3 proof] Convexity of G^{-1}_{Y_{1:n}}(F_{X_{1:n}}(x)) requires the second derivative of that composition to be nonnegative. The proof says it differentiates Eq. (4.8), but Eq. (4.8) is the ratio G^{-1}_{Y_{1:n}}(F_{X_{1:n}}(x))/x, not the composition itself. The displayed second-derivative expression is therefore not the relevant quantity for the convex transform order. Moreover, the claim that the expression is nonnegative under α*_k ≥ α_k is asserted without derivation. The theorem appears true, but the proof as printed is not valid.
  4. [Section 3, Theorem 3.3 proof] The displayed expression for the derivative of g_{Y_{1:n}}(x)/f_{X_{1:n}}(x) is not the derivative of that ratio. A direct calculation gives d/dx [g/f] = exp((A - A*)x) (A - A*) [ (A* + βx)(A + βx) + β ] / (A + βx)^2, where A = Σ α_k and A* = Σ α*_k, and this is nonnegative when A ≥ A*. The expression printed in the proof does not match this and does not by itself imply the likelihood ratio order. The theorem is true, but the proof as written contains an incorrect derivative.
minor comments (5)
  1. [Section 4, Eqs. (4.4)-(4.5)] The quantile of Y_{1:n} is written as F^{-1}_{Y_{1:n}} although the CDF of Y was denoted by G; use G^{-1}_{Y_{1:n}} for consistency.
  2. [Section 4, Eqs. (4.1)-(4.5)] The derivation uses the closure of the LFR family under minima of independent variables without stating it; a short lemma or remark would make the quantile derivation self-contained.
  3. [Section 5, Counterexample 5.6] The text says the plotted ratio is non-negative, but the relevant violation of the star order is that the ratio is not increasing; the caption should be reworded to state the actual monotonicity failure.
  4. [Introduction, literature review] The sentence beginning 'This paper discusses stochastic comparisons ... exponentiated Kumaraswamy-G distribution model' appears to summarize a different manuscript and should be corrected or removed.
  5. [Section 5, figures] The figures use the transformation x = log(1/(1-y)) but many captions label the horizontal axis as p or x; standardize the notation so that the plotted variable is unambiguous.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stochastic-order theorems are derived directly from the LFR definition, with self-citations confined to background and no fitted parameters.

full rationale

I walked the paper's derivation chain and found no load-bearing circular step. The theorems in Sections 3 and 4 are proved directly from the linear failure rate CDF (1.1), the independence assumption, and the standard stochastic-order definitions in Definitions 2.1 and 2.2; no parameter is fitted to a target ordering, and no result is imported from the authors' prior work as a premise. The quantile functions (4.4)-(4.5) are obtained by inverting the displayed CDF (3.1)-(3.2) algebraically, so the later dispersive, star, and convex-transform comparisons are genuine derived statements rather than restatements of the assumptions. The self-citation [18], which shares an author with the present paper, appears only in the introductory literature review and does not feed into any proof. The manuscript's Section 4 derivative displays, such as Eq. (4.6), appear to omit or misstate the required derivative factors, and the composition in Eq. (4.8) is not differentiated correctly in the proof of Theorem 4.2; these are verification/correctness concerns, not circularity, because the theorem statements are not assumed as inputs and the conditions are not constructed so that the conclusions hold by definition. No fitted-input-called-prediction, self-definitional equivalence, renaming of a known result, or author-imported uniqueness argument is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters or invented entities; the sufficient conditions are expressed directly in the LFR parameters. The model assumptions (independence, LFR form, standard stochastic order definitions) are the only external inputs. A self-citation to [18] is background only and does not enter any proof.

assumptions (4)
  • standard math Standard definitions of the usual stochastic, hazard rate, likelihood ratio, dispersive, star, convex, and Lorenz orders as given in Shaked and Shanthikumar.
    Used throughout Sections 2-4 as the framework for the comparison claims.
  • domain assumption The LFR distribution is absolutely continuous on R+ with CDF F(x)=1-exp(-(αx+βx^2/2)).
    This is the model under study, stated in Section 1, Eq. (1.1).
  • domain assumption Component lifetimes are independent and heterogeneous.
    Stated in Section 1 and used to write the series and parallel system survival and quantile functions as products and sums.
  • standard math The minimum of independent LFR variables is LFR with parameters (Σα_k, Σβ_k).
    Implicit in the quantile function derivation in Section 4, Eqs. (4.4)-(4.5); follows from the product form of the survival function but is never stated as a lemma.

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Cite this review

Pith. "Pith review of Ordering Results between Two Extreme Order Statistics with Heterogeneous Linear Failure Rate Distributed Components." pith.science (2026). https://pith.science/paper/QLRLBM66

@misc{pith2026250605773,
  author       = {Pith},
  title        = {Pith review of: Ordering Results between Two Extreme Order Statistics with Heterogeneous Linear Failure Rate Distributed Components},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QLRLBM66}},
  note         = {Machine review of arXiv:2506.05773}
}
read the original abstract

Stochastic comparisons of series and parallel systems are important in many areas of engineering, operations research and reliability analysis. These comparisons allow for the evaluation of the performance and reliability of systems under different conditions, and can inform decisions related to system design, probabilities of failure, maintenance and operation. In this paper, we investigate the stochastic comparisons of the series and parallel systems under the assumption that the component lifetimes have independent heterogeneous linear failure rate distributions. The comparisons are established based on the various stochastic orders including magnitude, transform and variability orders. Several numerical examples and counterexamples are constructed to illustrate the theoretical outcomes of this paper. Finally, we summarized our findings with a real-world application and possible future scopes of the present study.

Figures

Figures reproduced from arXiv: 2506.05773 by the authors.

Figure 1
Figure 1. (a) Plots of the difference G¯ Y1:3 (y) − F¯X1:3 (y) as in Example 5.1. (b) Plots of the difference hX1:3 (y) − hY1:3 (y) as in Example 5.2. The next counterexample shows the importance of the sufficient conditions “αk ≤ α ∗ k ” and “βk ≤ β ∗ k ” to establish the usual stochastic ordering between two series systems in Theorem 3.1. Counterexample 5.1 Assume αk ≤ α ∗ k and βk ≤ β ∗ k . Clearly, the assumptions made in… view at source ↗
Figure 2
Figure 2. Difference of the survival functions of Y1:3 and X1:3. (a): represents the difference between two survival functions is not always non-negative in y under the conditions αk < α ∗ k and βk > β∗ k . (b): represents the difference between two survival functions under the conditions αk > α∗ k and βk < β∗ k which is non-negative. (c): represents the curve of G¯ Y1:3 (y)− F¯X1:3 (y) under the conditions αk < α∗ k and βk <… view at source ↗
Figure 3
Figure 3. Plots represent the difference between two hazard rate functions of [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: (a) Plots of the ratio of the P DFs gY1:3 (y) and fX1:3 (y) as in Example 5.3. (b) Plot represents the ratio of the P DFs gY1:3 (y) and fX1:3 (y) as in Counterexample 5.3. We now present a counterexample to emphasize that the condition “αk ≥ α ∗ k ” is required for the…
Figure 5
Figure 5. Figure 5: (a) Plots of the difference G¯ Y3:3 (y) − F¯X3:3 (y) as in Example 5.4. (b) Plots of the difference G −1 Y1:3 (p) − F −1 X1:3 (p) as in Example 5.5. Next, present a counterexample to show that the sufficient conditions “αk ≥ α ∗ k ” and “βk ≥ β ∗ k ” are necessary for …
Figure 6
Figure 6. Figure 6: Plots represent the difference between two [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: (a) Plot of the difference G −1 Y1:3 (p) − F −1 X1:3 (p) as in Counterexample 5.5. (b) Plots of G −1 Y1:3 [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: (a) Plot of G −1 Y1:3 [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: Plot of G −1 Y1:3 [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

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Works this paper leans on

25 extracted references · 25 canonical work pages

  1. [1]

    An annotated overview of system-reliability opti- mization

    Way Kuo and V Rajendra Prasad. An annotated overview of system-reliability opti- mization. IEEE Transactions on Reliability , 49(2):176–187, 2000

    work page 2000

  2. [2]

    Order Statistics: An Introduction

    N Balakrishnan and Calyampudi R Rao. Order Statistics: An Introduction. Handbook of Statistics , 16:3–24, 1998

    work page 1998

  3. [3]

    Order Statistics

    Herbert Aron David and Haikady Navada Nagaraja. Order Statistics. Encyclopedia of Statistical Sciences, 2004

    work page 2004

  4. [4]

    Stochastic Orders

    Moshe Shaked and J George Shanthikumar. Stochastic Orders . Springer Science & Business Media, 2007

    work page 2007

  5. [5]

    Applications of likelihood ratio orderings in economics

    I Jewitt. Applications of likelihood ratio orderings in economics. Lecture Notes- Monograph Series, pages 174–189, 1991

    work page 1991

  6. [6]

    The hazard rate and the reversed hazard rate orders, with applications to order statistics

    Asok K Nanda and Moshe Shaked. The hazard rate and the reversed hazard rate orders, with applications to order statistics. Annals of the Institute of Statistical Mathematics , 53(4):853–864, 2001

    work page 2001

  7. [7]

    Stochastic comparisons of series and parallel systems with generalized exponential components

    Narayanaswamy Balakrishnan, Abedin Haidari, and Khaled Masoumifard. Stochastic comparisons of series and parallel systems with generalized exponential components. IEEE Transactions on Reliability , 64(1):333–348, 2014

    work page 2014

  8. [8]

    Stochastic comparisons in systems with Fr` echet distributed components.Operations Research Letters, 43(6):612– 615, 2015

    Nitin Gupta, Lakshmi Kanta Patra, and Somesh Kumar. Stochastic comparisons in systems with Fr` echet distributed components.Operations Research Letters, 43(6):612– 615, 2015

    work page 2015

Show all 25 references
  1. [9]

    Stochastic comparisons of parallel systems with exponentiated Weibull components

    Longxiang Fang and Xinsheng Zhang. Stochastic comparisons of parallel systems with exponentiated Weibull components. Statistics & Probability Letters, 97:25–31, 2015

    2015

  2. [10]

    Stochastic comparison of parallel systems with log-Lindley distributed components

    Shovan Chowdhury and Amarjit Kundu. Stochastic comparison of parallel systems with log-Lindley distributed components. Operations Research Letters, 45(3):199–205, 2017

    2017

  3. [11]

    New results on stochastic comparison of order statistics from heterogeneous Weibull populations

    Longxiang Fang and Xinsheng Zhang. New results on stochastic comparison of order statistics from heterogeneous Weibull populations. Journal of the Korean Statistical Society, 41(1):13–16, 2012

    2012

  4. [12]

    New results on stochastic comparisons of two- component series and parallel systems

    Neeraj Misra and Amit Kumar Misra. New results on stochastic comparisons of two- component series and parallel systems. Statistics & Probability Letters, 82(2):283–290, 2012

    2012

  5. [13]

    Stochastic order relations among parallel systems from Weibull distributions

    Nuria Torrado and Subhash C Kochar. Stochastic order relations among parallel systems from Weibull distributions. Journal of Applied Probability , 52(1):102–116, 2015

    2015

  6. [14]

    Ordering properties of order statistics from heterogeneous exponentiated Weibull models

    Amarjit Kundu and Shovan Chowdhury. Ordering properties of order statistics from heterogeneous exponentiated Weibull models. Statistics & Probability Letters, 114:119– 127, 2016. 19

    2016

  7. [15]

    Ordering of series and parallel systems comprising heterogeneous generalized modified Weibull com- ponents

    Narayanaswamy Balakrishnan, Phalguni Nanda, and Suchandan Kayal. Ordering of series and parallel systems comprising heterogeneous generalized modified Weibull com- ponents. Applied Stochastic Models in Business and Industry , 34(6):816–834, 2018

    2018

  8. [16]

    Some stochastic compar- ison results for series and parallel systems with heterogeneous Pareto type components

    Lakshmi Kanta Patra, Suchandan Kayal, and Phalguni Nanda. Some stochastic compar- ison results for series and parallel systems with heterogeneous Pareto type components. Applications of Mathematics , 63(1):55–77, 2018

    2018

  9. [17]

    Stochastic comparisons of parallel systems with generalized Kumaraswamy-G components

    Suchandan Kayal and Phalguni Nanda. Stochastic comparisons of parallel systems with generalized Kumaraswamy-G components. Communications in Statistics-Theory and Methods, 51(14):4712–4738, 2022

    2022

  10. [18]

    Some results on stochastic com- parisons of two finite mixture models with general components

    Suchandan Kayal, Raju Bhakta, and N Balakrishnan. Some results on stochastic com- parisons of two finite mixture models with general components. Stochastic Models , 39(2):363–382, 2023

    2023

  11. [19]

    Usual stochastic and reversed hazard orders of parallel systems with independent heterogeneous compo- nents

    Ghobad Barmalzan, Sajad Kosari, and Narayanaswamy Balakrishnan. Usual stochastic and reversed hazard orders of parallel systems with independent heterogeneous compo- nents. Communications in Statistics-Theory and Methods , 51(14):4781–4806, 2022

    2022

  12. [20]

    A new response time distribution

    D Kodlin. A new response time distribution. Biometrics, pages 227–239, 1967

    1967

  13. [21]

    Analysis for the linear failure-rate life-testing distribution

    Lee J Bain. Analysis for the linear failure-rate life-testing distribution. Technometrics, 16(4):551–559, 1974

    1974

  14. [22]

    Inference procedures for the linear failure rate model

    Ananda Sen and Gouri K Bhattacharyya. Inference procedures for the linear failure rate model. Journal of Statistical Planning and Inference , 46(1):59–76, 1995

    1995

  15. [23]

    Comparison methods for stochastic models and risks

    Alfred M¨ uller and Dietrich Stoyan. Comparison methods for stochastic models and risks. 2002

    2002

  16. [24]

    Stochastic orders in reliability and risk

    Haijun Li and Xiaohu Li. Stochastic orders in reliability and risk. Honor of Professor Moshe Shaked. Springer, New York , 2013

    2013

  17. [25]

    Variability ordering of heavy-tailed distributions with applications to order statistics

    Christian Kleiber. Variability ordering of heavy-tailed distributions with applications to order statistics. Statistics & Probability Letters, 58(4):381–388, 2002. 20

    2002

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