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arxiv: 1906.09937 · v1 · pith:C6IIIT5Wnew · submitted 2019-06-20 · 📊 stat.AP

Coherent systems with dependent and identically distributed components: A study of relative ageing based on cumulative hazard and cumulative reversed hazard rate functions

Nil Kamal Hazra , Neeraj Misra This is my paper

Pith reviewed 2026-05-25 19:21 UTC · model grok-4.3

classification 📊 stat.AP
keywords coherent systemsrelative ageingcumulative hazardcumulative reversed hazardk-out-of-n systemsstochastic ordersdependent components
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0 comments X

The pith

Sufficient conditions ensure one coherent system ages faster than another under cumulative hazard and reversed hazard orders when components are dependent and identically distributed.

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

The paper examines how coherent systems with dependent and identically distributed components age relative to each other. It supplies sufficient conditions on the component lifetimes that make one system age faster than another according to stochastic orders based on the cumulative hazard and cumulative reversed hazard rate functions. These conditions are shown to hold for k-out-of-n systems, with numerical examples confirming the comparisons.

Core claim

We give some sufficient conditions under which one coherent system ages faster than another one with respect to the ageing faster orders in the cumulative hazard and the cumulative reversed hazard rate functions. Further, we show that the proposed sufficient conditions are satisfied for k-out-of-n systems.

What carries the argument

Sufficient conditions on the component lifetime distributions that imply the ageing faster stochastic orders between the two coherent systems.

If this is right

  • The first system will be ordered ahead of the second in the relative ageing sense defined by the cumulative hazard function.
  • The same ordering will hold under the cumulative reversed hazard rate function.
  • All k-out-of-n systems satisfy the conditions and therefore exhibit the relative ageing property.
  • The comparisons remain valid for any dependence structure that preserves the identical distribution of components.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • System designers could apply the conditions to rank candidate designs by expected ageing speed.
  • The approach may extend to other coherent structures such as series-parallel combinations if analogous lifetime conditions can be checked.
  • Numerical verification for specific dependence models would strengthen the practical use of the conditions.

Load-bearing premise

The components of the coherent systems are dependent and identically distributed.

What would settle it

Two coherent systems with dependent identical components that meet the sufficient conditions yet fail to satisfy one ageing faster order relative to the other.

read the original abstract

The relative ageing is an important notion which is useful to measure how a system ages relative to another one. Among all existing stochastic orders, there are two important orders describing the relative ageing of two systems, namely, ageing faster orders in the cumulative hazard and the cumulative reversed hazard rate functions. In this paper, we give some sufficient conditions under which one coherent system ages faster than another one with respect to the aforementioned stochastic orders. Further, we show that the proposed sufficient conditions are satisfied for $k$-out-of-$n$ systems. Moreover, some numerical examples are given to illustrate the developed results.

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

0 major / 3 minor

Summary. The paper develops sufficient conditions on the lifetime distributions of dependent and identically distributed (DID) components under which one coherent system ages faster than another with respect to the cumulative-hazard and cumulative-reversed-hazard ageing orders. It then verifies that these conditions hold for k-out-of-n systems and supplies numerical examples.

Significance. The results extend relative-ageing comparisons to the DID setting that is common in reliability modeling when component lifetimes are exchangeable but dependent. Verification for the k-out-of-n family supplies a concrete, immediately usable class of systems.

minor comments (3)
  1. [Introduction] The abstract states that the proposed conditions are satisfied for k-out-of-n systems, but the introduction does not preview the precise form of the dependence structure (e.g., copula family or exchangeability assumptions) that is maintained throughout the derivations.
  2. Notation for the two ageing orders (cumulative hazard and cumulative reversed hazard) is introduced only after the statement of the main theorems; an early, self-contained definition would improve readability.
  3. [Numerical examples] The numerical examples section would benefit from an explicit statement of the joint distribution (or copula) used to generate the DID samples, together with the sample size or simulation method.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, recognition of the significance of extending relative-ageing results to the DID setting, and the recommendation of minor revision. The referee's description of the contributions is accurate.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from stochastic order assumptions

full rationale

The paper states sufficient conditions on component lifetime distributions for one coherent system to age faster than another under cumulative hazard and cumulative reversed hazard ageing orders, then verifies the conditions hold for k-out-of-n systems under the explicit DID premise. No load-bearing step reduces by construction to a fit, self-definition, or self-citation chain; all steps are forward derivations from the stated joint distribution assumptions and stochastic order definitions. The DID setup is the modeling premise, not a derived output. This matches the default expectation of non-circularity for mathematical papers on stochastic orders.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be identified from the abstract alone; the work appears to rest on standard properties of stochastic orders and coherent systems.

pith-pipeline@v0.9.0 · 5633 in / 1045 out tokens · 61353 ms · 2026-05-25T19:21:10.766335+00:00 · methodology

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Reference graph

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