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arxiv: 2606.06161 · v1 · pith:RW43IPMWnew · submitted 2026-06-04 · 🧮 math.ST · stat.TH

Monitoring the Ratio of two Normal Variables using EWMA Type Control Charts in Short Production Runs

Thi Hien Nguyen , Jean-Michel Masereel , Guillaume Tartare , Kim Duc Tran This is my paper

Pith reviewed 2026-06-27 23:24 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords EWMA control chartratio of normal variablesshort production runstruncated average run lengthMarkov chainstatistical process controlShewhart chart comparison
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The pith

An EWMA control chart for ratios of two normals detects small shifts faster than Shewhart charts in short production runs.

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

This paper develops an exponentially weighted moving average chart to monitor the ratio of two normally distributed quality characteristics when only a short sequence of inspections is available. It adopts the corrected closed-form density for the ratio and uses a Markov-chain model of the EWMA recursion to set control limits that achieve a target in-control truncated average run length over a finite horizon. A large numerical study shows the new chart responds more quickly to small and moderate shifts than the existing Shewhart-type short-run ratio chart, with the largest gains when the shift magnitude is five percent or less. The approach is demonstrated on a beverage filling process to show how the limits are computed and applied in practice.

Core claim

The proposed EWMA-RZ chart, calibrated via Markov chain to a prescribed TARL0 over finite horizon I using Nadarajah's corrected density, provides substantially better detection of small and moderate shifts than the ShRZ chart, especially for |τ - 1| ≤ 0.05.

What carries the argument

Markov-chain representation of the EWMA recursion on the ratio statistic that computes finite-horizon TARL0 and out-of-control TARL1 values from the corrected closed-form density of the ratio of two normals.

If this is right

  • The EWMA-RZ chart outperforms ShRZ across variations in smoothing constant, in-control correlation, coefficients of variation, sample size, and shift size.
  • The detection advantage is largest for the smallest shifts examined.
  • The method applies directly to processes such as beverage filling where performance is tracked by a ratio.
  • Control limits are obtained once per parameter set and then applied to successive ratio observations within the finite horizon.

Where Pith is reading between the lines

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

  • The same Markov-chain calibration could be reused for other memory-type charts on ratios if the density remains valid.
  • In industries limited to short batches, adopting this chart would reduce the average number of inspections needed to signal a small drift in the ratio.
  • If the normality or independence assumptions on the underlying variables are mildly violated, the relative ranking versus Shewhart may still hold provided the ratio density approximation remains reasonable.

Load-bearing premise

The ratio of the two normal variables follows the corrected closed-form density of Nadarajah (2020) and the Markov-chain representation accurately captures the finite-horizon TARL0 behavior of the EWMA recursion.

What would settle it

A Monte Carlo simulation of the ratio process under the stated density where the out-of-control TARL1 of the EWMA-RZ chart is not lower than that of the ShRZ chart for |τ - 1| ≤ 0.05 would falsify the performance claim.

Figures

Figures reproduced from arXiv: 2606.06161 by Guillaume Tartare, Jean-Michel Masereel, Kim Duc Tran, Thi Hien Nguyen.

Figure 1
Figure 1. Figure 1: Detection profile of the EWMA-RZ+ chart: TARL1 as a function of the shift factor τ , for the equal-CV case (γX, γY ) = (0.2, 0.2), ρ0 = 0.4, λ = 0.2, I = 10, TARL∗ 0 = 10. 5.5 Comparison with the ShRZ chart of Tran et al.22 For a like-for-like comparison with the Shewhart-type ShRZ chart of Tran et al. 22, we use the same parameter grid and the same target TARL∗ 0 = I. Across the upward-shift levels τ ∈ {1… view at source ↗
Figure 2
Figure 2. Figure 2: Illustrative EWMA-RZ+ chart for the beverage-filling example. The process drifts upward to τ = 1.05 at inspection i = 9 and the chart signals at i = 10. Calibrated UCL = 1.01918, z0 = 1, γX = γY = 0.05, ρ0 = 0.4, λ = 0.2, n = 5, I = 20, TARL∗ 0 = 20. For comparison, the ShRZ Shewhart-type chart of Tran et al. 22 applied to the same simulated run signals only later: a single Zˆ i at the post-shift level τz0… view at source ↗
read the original abstract

In many industrial and engineering applications, process performance is characterized by the ratio of two normally distributed quality characteristics. Monitoring such ratios is particularly challenging in short production runs, where conventional control charts often suffer from limited sensitivity due to the small number of available inspections. This paper proposes an exponentially weighted moving average (EWMA) control chart for monitoring the ratio of two normally distributed random variables under short production run (SPR) conditions. The statistical distribution of the ratio is first reviewed, adopting the corrected closed-form density of Nadarajah (2020) rather than the approximation used in earlier studies. The control limit of the proposed chart is calibrated to a prescribed in-control truncated average run length (TARL$ _0 $) over a finite horizon $ I $ of inspections, using a Markov-chain representation of the EWMA recursion. The detection performance of the chart is then assessed through a large factorial study covering the smoothing constant $ \lambda $, the in-control correlation $ \rho_0 $, the coefficients of variation $ (\gamma_X, \gamma_Y) $, the sample size $ n $, and the magnitude of the shift $ \tau $. Numerical results show that the proposed EWMA-RZ chart provides substantially better detection of small and moderate shifts than the recently developed Shewhart-type short-run ratio chart (ShRZ) of Tran et al. (2021), especially for $ |\tau - 1| \le 0.05 $. An illustrative example based on a beverage filling process is included to demonstrate the practical implementation of the method.

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

2 major / 2 minor

Summary. The paper proposes an EWMA control chart (EWMA-RZ) for monitoring the ratio of two normally distributed variables in short production runs. It reviews the ratio distribution using Nadarajah's (2020) corrected density, calibrates the control limits to a target in-control truncated average run length (TARL0) over a finite horizon using a Markov-chain representation of the EWMA recursion, and evaluates performance via a factorial simulation study. The results indicate that the proposed chart detects small and moderate shifts in the ratio parameter τ more effectively than the Shewhart-type ShRZ chart of Tran et al. (2021), particularly for |τ - 1| ≤ 0.05. An illustrative example is included.

Significance. Should the central numerical comparisons hold, this work contributes a more sensitive monitoring tool for ratio-based quality characteristics in short-run production environments, which are common in manufacturing. The adoption of the exact density and the comprehensive factorial design over parameters such as λ, ρ0, γX, γY, n, and τ provide a solid basis for the performance assessment. The direct comparison to an existing short-run chart is a strength.

major comments (2)
  1. [Markov-chain approach section] The TARL0 calibration relies on the Markov-chain discretization of the EWMA recursion (described in the section on the proposed chart), but no direct Monte Carlo simulation is reported to verify the accuracy of the transition matrix and absorbing-state handling for the finite horizon I. This verification is necessary to support the in-control calibration and the out-of-control TARL1 comparisons that underpin the superiority claim over ShRZ for |τ-1|≤0.05.
  2. [Numerical results section] Table or figure presenting the factorial simulation results: the reported performance gains for small shifts should include measures of variability (e.g., standard errors on estimated TARL1 values) to allow assessment of whether the differences from the ShRZ benchmark are statistically meaningful across the design points.
minor comments (2)
  1. [Abstract] The abstract introduces TARL0 without a brief definition; adding a parenthetical note on its meaning as truncated average run length would improve accessibility.
  2. [Introduction or model section] Notation for the shift parameter τ and the in-control correlation ρ0 should be consistently defined at first use in the main text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below, indicating planned revisions where appropriate to strengthen the presentation of the Markov-chain calibration and the simulation results.

read point-by-point responses

  1. Referee: [Markov-chain approach section] The TARL0 calibration relies on the Markov-chain discretization of the EWMA recursion (described in the section on the proposed chart), but no direct Monte Carlo simulation is reported to verify the accuracy of the transition matrix and absorbing-state handling for the finite horizon I. This verification is necessary to support the in-control calibration and the out-of-control TARL1 comparisons that underpin the superiority claim over ShRZ for |τ-1|≤0.05.

    Authors: We agree that an explicit verification step would increase confidence in the finite-horizon Markov-chain implementation. Although the discretization approach follows standard methodology (e.g., Lucas & Saccucci 1990 and subsequent short-run extensions), we will add a brief verification subsection that reports a side-by-side comparison of Markov-chain TARL0/TARL1 values against direct Monte Carlo simulation for a representative subset of design points (λ, ρ0, γX, γY, I). This will be included in the revised manuscript. revision: yes

  2. Referee: [Numerical results section] Table or figure presenting the factorial simulation results: the reported performance gains for small shifts should include measures of variability (e.g., standard errors on estimated TARL1 values) to allow assessment of whether the differences from the ShRZ benchmark are statistically meaningful across the design points.

    Authors: We acknowledge that reporting variability measures would allow readers to judge the precision of the reported differences. The TARL1 values in the factorial study are obtained from Monte Carlo simulation (with a fixed large number of replications). In the revision we will augment the main tables with standard errors (or 95% intervals) for the estimated TARL1 entries, computed from the simulation replicates, so that the statistical significance of the gains relative to ShRZ can be assessed directly. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to comparator chart; derivation remains independent

full rationale

The paper proposes an EWMA-RZ chart, adopts the external Nadarajah (2020) closed-form density for the ratio, and calibrates control limits to a target TARL0 via Markov-chain discretization of the EWMA recursion over finite horizon I. Performance is then evaluated via TARL1 in a factorial design and compared to the ShRZ chart from Tran et al. (2021). This comparison involves a self-citation (overlapping author Kim Duc Tran), but the central claim of superior detection for small shifts is an empirical numerical result under the adopted model, not a reduction of any prediction or limit to a fitted input by construction. No self-definitional steps, no fitted parameters renamed as predictions, no uniqueness theorems imported from the authors' prior work, and no ansatz smuggled via citation. The Markov-chain approach is a standard approximation technique whose accuracy is an external verification issue, not a circularity issue. This qualifies as at most minor self-citation that is not load-bearing for the derivation chain.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Relies on standard normality assumption for the two variables and an external density formula; no new entities postulated.

free parameters (1)
  • lambda
    Smoothing constant varied in the factorial study to assess performance.
axioms (2)
  • domain assumption X and Y are independent or correlated normal random variables.
    Stated as quality characteristics whose ratio is monitored.
  • domain assumption The corrected closed-form density of Nadarajah (2020) is accurate for the ratio.
    Adopted explicitly instead of earlier approximations.

pith-pipeline@v0.9.1-grok · 5847 in / 1193 out tokens · 47804 ms · 2026-06-27T23:24:27.380663+00:00 · methodology

discussion (0)

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Reference graph

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