REVIEW 3 major objections 6 minor 3 references
Stochastic Comparisons of Series and Parallel Systems with Topp-Leone Generated Family of Distributions
T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read In the Topp-Leone generated family, majorization of shape parameters orders series-system hazard rates and weak submajorization of scale parameters orders parallel-system lifetimes.
desk verdict Incremental but correct extension of majorization orderings to the Topp-Leone-G family; the central theorems hold up, but Theorem 3.5 is unreadable and needs a rewrite. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the representation of a TL-G series-system hazard rate as $2\theta g(x;\xi)G(x;\xi)^{\theta-1}(1-G(x;\xi)^\theta)\sum_{k=1}^n\tau(\alpha_k)$, where $\tau(\alpha)=\alpha t^{\alpha-1}/(1-t^\alpha)$ and $t=G(x;\xi)^\theta(2-G(x;\xi)^\theta)\in(0,1)$. Since $\tau$ is convex in $\alpha$ (Lemma 2.3), the sum is Schur-convex by Lemma 2.4, so majorization of $\alpha$ orders the system hazard rates. For parallel systems the machinery is the system distribution function $\varphi(\theta)=\prod_{k=1}^n\left(G(x;\xi)^{\theta_k}(2-G(x;\xi)^{\theta_k})\right)^\alpha$, which is shown to be symmetric, decreasing in each coordinate, and Schur-concave; negating it converts weak submajorization of the scale vector into the usual stochastic order.
What would settle it
A reader can settle the main series-system claim by direct calculation: check whether $\tau''(\alpha)\ge 0$ for all $\alpha>0$ at several values of $t$ in $(0,1)$; any negative value would invalidate Lemma 2.3 and with it the proof of Theorem 3.1. Alternatively, for a chosen baseline such as $G(x;\xi)=1-e^{-x}$, compute the two series-system hazard rates for shape vectors $\alpha=(1,9)$ and $\alpha^*=(4,6)$ and search over $x$ for a point where the first hazard rate is smaller than the second, which would contradict the theorem's conclusion.
Extended reading notes
Core claim
The central claim is that for the TL-G family the reliability comparison of series and parallel systems reduces to a comparison of the parameter vectors by majorization. The main series-system result is Theorem 3.1: with fixed scale $\theta>0$ and fixed baseline $G$, if $\alpha^*$ is majorized by $\alpha$, then $X_{1:n}$ with shape vector $\alpha$ is smaller than $Y_{1:n}$ with shape vector $\alpha^*$ in the hazard rate order. The parallel-system results are Theorem 3.2 and Theorem 3.3: $\theta \prec_w \theta^*$ (or $\theta_k\le \theta_k^*$ for every $k$) implies $X_{n:n}\le_{st}Y_{n:n}$; Theorem 3.4 characterizes the likelihood ratio order between parallel systems by $\sum_{k=1}^n\alpha_k\le\sum_{k=1}^n\alpha_k^*$, and Theorem 3.5 and 3.6 transfer these orderings to systems whose component baselines differ when the baselines themselves are stochastically ordered.
Load-bearing premise
The load-bearing premise is the imported Lemma 2.3, which says the function $\tau(\alpha)=\alpha t^{\alpha-1}/(1-t^\alpha)$ is convex in $\alpha$ for every fixed $t$ in $(0,1)$; if that convexity failed for even one $t$, the proof of the main series-system hazard-rate ordering would no longer go through.
Editorial extensions
If this is right
- Series systems with the same scale and baseline but more heterogeneous shape parameters are less reliable in the hazard rate sense, so a designer can order two designs directly from the majorization relation between their shape vectors.
- Parallel systems with scale vectors that are larger in the weak submajorization sense are stochastically longer-lived, covering configurations in which no single component scale dominates across the board.
- When only shape parameters vary, the likelihood ratio order between two parallel systems is fully decided by the sums of the shape parameters, giving a one-number rule for this comparison.
- The results persist when the two systems are built on different baseline distributions, provided the baselines are stochastically ordered and the relevant parameter vectors satisfy the majorization conditions.
- The counterexamples in the paper mark the boundary: the hazard-rate and usual-stochastic-order results for series and parallel systems do not upgrade to likelihood ratio order.
Reading between the lines
- A direct proof of the convexity of $\tau$ would make the main series-system argument self-contained, and the same inequality is the natural quantity to check when carrying the result to other generated families.
- The same factorized-hazard-rate structure suggests the series-system theorem may extend to any generated family for which the corresponding $\tau$ is convex, not only TL-G; testing that would require only substituting another baseline generator into the proof.
- Because Theorem 3.4 is an if-and-only-if statement about sums of shape parameters, it implies that for parallel systems the shape parameters affect likelihood ratio ordering only through their total, a feature worth testing for other generated families.
- An unstated practical reading of the weak submajorization result is that equalizing scale parameters across components tends to shorten parallel-system lifetimes, so redundancy designs should concentrate scale increases on the larger-scale components; this follows from the paper's monotonicity proof but the paper does not draw it out.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies stochastic comparisons of series and parallel systems whose component lifetimes follow the Topp-Leone generated (TL-G) family of distributions. It establishes: (i) hazard-rate ordering of series systems when the shape-parameter vector is majorized (Theorem 3.1); (ii) usual stochastic ordering of parallel systems when the θ-parameter vectors are weakly submajorized or componentwise ordered (Theorems 3.2, 3.3 and Corollary 3.1); (iii) likelihood-ratio ordering of parallel systems if and only if the sums of shape parameters are ordered (Theorem 3.4); and (iv) extensions to comparisons of systems with different baseline distributions (Theorems 3.5 and 3.6). The proofs use standard majorization and Schur-convexity arguments, with numerical examples and counterexamples showing that likelihood-ratio versions of Theorems 3.1 and 3.2 fail.
Significance. If the results are correct, they provide valid ordering theorems for a recently introduced lifetime family and extend the existing catalogue of majorization-based comparisons for series and parallel systems. The central derivations are standard and appear mathematically sound: the hazard-rate decomposition in Theorem 3.1, the Schur-concavity argument in Theorem 3.2, and the likelihood-ratio equivalence in Theorem 3.4 are all consistent with the stated distributional framework. The paper also provides explicit counterexamples that usefully demarcate the limits of the ordering results. The contribution is incremental, but it is within the scope of the journal and would be a useful reference for the reliability community provided the presentation issues below are fixed.
major comments (3)
- [Theorem 3.5] The statement of Theorem 3.5 is not readable as typeset; the string "x: Sgt}? 2:? Lhnifmtvfim-" cannot be parsed as a mathematical condition, and the proof contains the impossible assumption "X* ≤_st X*". It is therefore impossible to verify the hypotheses, in particular whether U_k has shape parameter α_k and V_k has shape parameter α_k^*, and what stochastic order is assumed between the baseline variables with CDFs G_1 and G_2. Please restate Theorem 3.5 and its proof in full, clear notation; as it stands, this main result cannot be checked.
- [Eq. (1.1)] As displayed, the density in Eq. (1.1) is inconsistent with the CDF in Eq. (1.2). Differentiating F(x) = (G(x;ξ)^θ(2 - G(x;ξ)^θ))^α gives 2αθg(x;ξ)G(x;ξ)^{θ-1}(1 - G(x;ξ)^θ)(G(x;ξ)^θ(2 - G(x;ξ)^θ))^{α-1}, which contains an additional factor (G(x;ξ)^θ)^{α-1} relative to the printed formula. The proof of Theorem 3.1 implicitly uses the correct density through τ(α) = αt^{α-1}/(1-t^α) with t = G^θ(2-G^θ). Please correct Eq. (1.1) or explicitly state the intended definition of the TL-G density.
- [Lemma 2.3] Lemma 2.3 is the single non-obvious ingredient in the proof of Theorem 3.1, since the convexity of τ(α) = αt^{α-1}/(1-t^α) is exactly what makes the sum ∑τ(α_k) Schur-convex. The manuscript cites Balakrishnan et al. (2014) without proof or a precise lemma number. Please include a proof, or at least an exact quotation, of this lemma so that Theorem 3.1 is verifiable from the paper itself.
minor comments (6)
- [Corollary 3.1 and Theorem 3.3] Corollary 3.1 and Theorem 3.3 state essentially the same componentwise-ordering result; please merge them or explain the intended distinction.
- [Throughout] The index 'h=1,2,...,n' consistently appears where 'k=1,2,...,n' is meant; for example, in the statements of Theorems 3.1, 3.2, and 3.4 this is a repeated notational slip.
- [Examples 3.1 and 3.2] In Examples 3.1 and 3.2 the second sample is denoted 'Y2' in the text but should be 'Y_k' for k=1,2; please correct the notation.
- [Theorem 3.6] The notation X_f and X_g is introduced only in the preamble and is not defined in the theorem statement; please define the baseline random variables explicitly in Theorem 3.6.
- [Lemmas 2.1-2.4] The reference years in Lemmas 2.1-2.4 are typeset as '2611' and '2614'; these should read 2011 and 2014, respectively.
- [Figures 1 and 2] The captions for Figures 1 and 2 are missing; the plots are referenced only inside the text, so please add complete captions describing what is displayed.
Circularity Check
No circularity: the ordering theorems follow from the TL-G definitions and external majorization/convexity lemmas.
full rationale
The paper makes no circular steps. The main series-system hazard-rate ordering (Theorem 3.1) is derived from the TL-G density and distribution functions (1.1)-(1.2), the series-system hazard identity, an algebraic rewrite to the sum tau(alpha_k) with t = G(x; E)^theta (2 - G(x; E)^theta), and the cited convexity of tau (Lemma 2.3, from Balakrishnan et al. 2014, with no author overlap) together with the Schur-convex sum lemma (Lemma 2.4). Theorems 3.2 and 3.3 derive the usual stochastic order by direct differentiation of the parallel-system c.d.f. (3.1) and by Schur-concavity or coordinatewise monotonicity; Theorem 3.4 gives an iff by direct monotonicity of the likelihood ratio. Theorems 3.5 and 3.6 chain previously proved results rather than assuming their conclusions. Nothing is fitted, no parameter is renamed as a prediction, and no theorem assumes the ordering it claims to prove. The load-bearing external ingredients are standard majorization facts or a known convexity result whose stated assumptions do not include the target ordering. The only weakness is that Lemma 2.3 is quoted rather than proved in this paper, but that is an external citation, not a self-citation chain, and the lemma is independently checkable; it does not make the derivation circular.
Assumptions & free parameters
assumptions (4)
- standard math Lemma 2.3: tau(alpha)=alpha t^{alpha-1}/(1-t^alpha) is convex in alpha for each t in (0,1), cited from Balakrishnan et al. (2014).
- standard math Schur-concavity and weak submajorization criteria from Marshall et al. (2011), specifically Lemmas 2.1 and 2.2.
- domain assumption The TL-G density and CDF as defined by Rezaei et al. (2017) in equations (1.1) and (1.2).
- domain assumption Independence of component lifetimes and common (or stochastically ordered) baseline distributions G1 and G2 in the theorems.
Cite this review
Pith. "Pith review of Stochastic Comparisons of Series and Parallel Systems with Topp-Leone Generated Family of Distributions." pith.science (2026). https://pith.science/paper/2BM6P3PY
@misc{pith2026190805896,
author = {Pith},
title = {Pith review of: Stochastic Comparisons of Series and Parallel Systems with Topp-Leone Generated Family of Distributions},
year = {2026},
howpublished = {\url{https://pith.science/paper/2BM6P3PY}},
note = {Machine review of arXiv:1908.05896}
}
read the original abstract
In this article, we stochastically compare the series and parallel systems having Topp Leone generated family of distributions. We consider that the lifetimes of the components of the systems have either the different shape parameters when the scale parameters are fixed or the different scale parameters when the shape parameters are fixed and established some ordering results With the help of vector majorization technique.
Reference graph
Works this paper leans on
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[1]
BALAKRISHNAN, N., HAIDARI, A. AND MASOUMIFARD, K. (2014). Stochastic com— parisons of series and parallel systems with generalized exponential components. IEEE Transactions on Reliability, 64, 333—348. BALAKRISHNAN, N. AND RAO, C. R. (1998). Order statistics: Applications. Handbook of Statistics, Amsterdam. BALAKRISHNAN, N. AND ZHAO, P. (2013). Hazard rat...
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BARMALZAN, G., NAJAFABADI, A. T. P. AND BALAKRISHNAN, N. (2016). Like— lihood ratio and dispersive orders for smallest order statistics and smallest claim amounts from heterogeneous Weibull sample. Statistics and Probability Letters, 110, 1:7. DAVID, H. A. AND NAGARAJA, H. N. (2003). Order statistics, 3rd edition. Wiley Series in Probability and Statistic...
work page 2016
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NADARAJAH, 8., JIANG, X. AND CHU, J. (2017). Comparisons of smallest order statistics from Pareto distributions with different scale and shape parameters. Annals of Operations Research, 254, 191:209. PATRA, L. K., KAYAL, S. AND NANDA, P. (2018). Some stochastic comparison results for series and parallel systems with heterogeneous Pareto type components. A...
work page 2017
Reviewed August 14, 2026 · model on record in the stance chip above.
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