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arxiv: 2606.06137 · v1 · pith:LCC3WR3Xnew · submitted 2026-06-04 · 🧮 math.ST · stat.TH

An Adaptive Upper One-Sided Cumulative Sum Control Chart with Joint Parameter Optimization for Monitoring the Ratio of Two Normal Variables in Short Production Runs

Kim Duc Tran This is my paper

Pith reviewed 2026-06-27 23:28 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords CUSUM control chartratio monitoringshort production runsadaptive parameter optimizationMarkov chain approximationtruncated average run lengthbivariate normal variables
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The pith

An adaptive CUSUM chart for the ratio of two normal variables jointly optimizes its reference value and decision interval to meet a target in-control run length while lowering out-of-control run length in short production runs.

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

The paper proposes an upper one-sided CUSUM control chart for monitoring the ratio Z equals X over Y of two correlated normal variables during short production runs. It replaces fixed reference-value designs with a bilevel procedure that tunes both the reference value k and the decision interval h to hit a prescribed in-control truncated average run length while minimizing the out-of-control value. The inner step solves for h given k via root-finding; the outer step searches over k to minimize the out-of-control TARL1. Both steps rely on a finite-state Markov-chain model that incorporates a ratio approximation. The resulting adaptive chart is compared with Shewhart, EWMA, and fixed-k CUSUM alternatives on matched horizons and under correlation shifts between phases.

Core claim

The adaptive CUSUM-RZ+ chart with joint optimization of k and h satisfies the target TARL0 equals I while achieving lower out-of-control TARL1 than fixed-k CUSUM-RZ+, Shewhart-RZ, and EWMA-RZ charts, and improves when correlation increases from Phase I to Phase II.

What carries the argument

Bilevel optimization that calibrates the decision interval h(k) by inner root-finding to enforce the TARL0 constraint and selects the optimal reference value k* by outer line search to minimize TARL1, all evaluated inside a finite-state Markov-chain framework with ratio approximation.

If this is right

  • All memory-type charts outperform the Shewhart-RZ baseline under the same horizon.
  • The adaptive design matches other memory charts when correlation stays stable across phases.
  • Performance improves when correlation rises from Phase I to Phase II.
  • The chart remains insensitive to symmetric heavy tails but becomes mildly anti-conservative under contamination.
  • At least 100 Phase I subgroups keep relative bias in TARL0 near one.

Where Pith is reading between the lines

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

  • The same bilevel search could be applied to two-sided or lower-sided versions of the ratio chart.
  • The Markov-chain framework might be reused for other ratio-based statistics in finite-horizon settings.
  • The method suggests that joint parameter search can recover feasible designs even when single-parameter tuning hits boundary solutions.

Load-bearing premise

The finite-state Markov-chain model together with the ratio approximation correctly reproduces the run-length distribution of the CUSUM statistic for the ratio Z equals X over Y under bivariate normality.

What would settle it

Monte Carlo simulation of the actual TARL0 under the optimized k and h values that shows large deviation from the target I would falsify the claim that the bilevel procedure meets the in-control constraint.

Figures

Figures reproduced from arXiv: 2606.06137 by Kim Duc Tran.

Figure 1
Figure 1. Figure 1: Diagnostic of the bilevel adaptive search for [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: TARL1 of the adaptive CUSUM-RZ+ chart on a log scale, equal CV (γX, γY ) = (0.20, 0.20), ρ1 = ρ0, τ0 = I = 30. 1.00 1.02 1.04 1.06 1.08 1.10 10 1 6 × 10 0 2 × 10 1 3 × 10 1 T A R L1 ( X, Y) = (0.01, 0.20) 0 = 1 = 0.0 1.00 1.02 1.04 1.06 1.08 1.10 10 1 6 × 10 0 2 × 10 1 3 × 10 1 ( X, Y) = (0.01, 0.20) 0 = 1 = 0.4 I = TARL0 = 30, unequal CV, 1 = 0 n = 5 n = 10 [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: TARL1 of the adaptive CUSUM-RZ+ chart on a log scale, unequal CV (γX, γY ) = (0.01, 0.20), ρ1 = ρ0, τ0 = I = 30. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: TARL1 of the adaptive CUSUM-RZ+ chart on a log scale, equal CV (γX, γY ) = (0.20, 0.20), ρ0 = 0.4, ρ1 = 0.8, τ0 = I = 30. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: TARL1 of the adaptive CUSUM-RZ+ chart on a log scale, unequal CV (γX, γY ) = (0.01, 0.20), ρ0 = 0.4, ρ1 = 0.8, τ0 = I = 30. 6.3 Adaptive vs. fixed-k design To quantify the benefit of joint (k, h) optimization over the fixed-k design, Ta￾ble 10 compares the two procedures under matched in-control TARL0 = 30 for the parameter combinations of Tables 4–5. For each row, the fixed-k design uses kfixed = 1.025 an… view at source ↗
Figure 6
Figure 6. Figure 6: Matched-horizon benchmark of the four short-run ratio charts for [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Monte Carlo TARL of the adaptive CUSUM-RZ [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Phase I estimation effect on the achieved TARL [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗
read the original abstract

Monitoring the ratio of two correlated normal variables is increasingly important in statistical process control, since many quality characteristics are expressed in relative rather than absolute form. Memory-type ratio charts have mostly been developed for long production runs, while their finite-horizon counterparts rely on a fixed reference value $ k $ derived from a specified shift. Such fixed-$ k $ designs are not optimal at a given out-of-control magnitude and, in low-variability regimes, yield boundary solutions for which the in-control truncated average run length (TARL$ _0 $) is unattainable. This paper proposes an upper one-sided cumulative sum (CUSUM) control chart for the ratio $ Z = X/Y $ in short production runs, denoted CUSUM-RZ$ ^+ $ (RZ standing for the ratio $ Z $), with fully adaptive joint optimization of $ k $ and the decision interval $ h $. Given a target TARL$ _0 = I $ and a target shift $ \tau $, a bilevel problem calibrates $ h(k) $ by inner root-finding to satisfy the TARL$ _0 $ constraint and selects $ k^* $ by outer line search to minimize the out-of-control TARL$ _1 $. Both use a finite-state Markov-chain framework with an accurate ratio approximation; the inner step recovers boundary cases that fixed-$ k $ designs cannot. The chart is assessed through matched-horizon benchmarks against Shewhart-RZ, exponentially weighted moving average (EWMA-RZ), and fixed-$ k $ CUSUM-RZ$ ^+ $ charts, Monte Carlo robustness studies, and a Phase I estimation analysis. All memory-type charts outperform the Shewhart-RZ baseline; the adaptive design matches them under stable correlation and improves appreciably when correlation rises from Phase I to Phase II. It is insensitive to symmetric heavy tails yet mildly anti-conservative under contamination, and $ m \geq 100 $ subgroups keep the TARL$ _0 $ relative bias near 1%.

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

1 major / 0 minor

Summary. The paper proposes an adaptive upper one-sided CUSUM chart (CUSUM-RZ+) for monitoring the ratio Z = X/Y of two correlated normal variables in short production runs. It formulates a bilevel optimization that, for a target TARL0 = I and shift τ, solves an inner root-finding problem for the decision interval h(k) via a finite-state Markov-chain model with ratio approximation, then performs an outer line search over k to minimize TARL1. The resulting adaptive design is benchmarked against Shewhart-RZ, EWMA-RZ, and fixed-k CUSUM-RZ+ charts on matched horizons, with additional Monte Carlo robustness checks and Phase-I estimation analysis; it is reported to recover boundary cases unattainable by fixed-k designs and to improve when correlation increases from Phase I to Phase II.

Significance. If the Markov-chain representation and ratio approximation are accurate, the joint optimization supplies a practical procedure that satisfies the in-control TARL0 constraint while delivering lower out-of-control TARL1 than fixed-parameter alternatives, particularly under correlation drift. The explicit recovery of boundary solutions and the use of the Markov chain to enforce the TARL0 = I constraint are concrete strengths that address a known limitation of memory-type ratio charts in finite-horizon settings.

major comments (1)
  1. [Abstract and bilevel-problem description] Abstract and the bilevel-problem description (paragraph beginning 'Given a target TARL0 = I'): the central performance claims rest on the finite-state Markov-chain framework together with the ratio approximation correctly yielding the TARL0 and TARL1 values that drive both the inner root-finding for h(k) and the outer search for k*. No derivation of the transition probabilities, no error analysis or bound on the ratio approximation, and no verification that the inner root-finding converges for all admissible k are supplied; these omissions are load-bearing because the reported superiority over fixed-k CUSUM-RZ+ and the other benchmarks is obtained directly from the TARL values produced by this model.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback and for emphasizing the importance of methodological transparency in the bilevel optimization framework. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract and bilevel-problem description] Abstract and the bilevel-problem description (paragraph beginning 'Given a target TARL0 = I'): the central performance claims rest on the finite-state Markov-chain framework together with the ratio approximation correctly yielding the TARL0 and TARL1 values that drive both the inner root-finding for h(k) and the outer search for k*. No derivation of the transition probabilities, no error analysis or bound on the ratio approximation, and no verification that the inner root-finding converges for all admissible k are supplied; these omissions are load-bearing because the reported superiority over fixed-k CUSUM-RZ+ and the other benchmarks is obtained directly from the TARL values produced by this model.

    Authors: We agree that the manuscript would benefit from greater explicit detail on the Markov-chain construction and ratio approximation to make the performance claims fully reproducible. In the revision we will insert a dedicated subsection that (i) derives the transition probabilities of the finite-state Markov chain from the distribution of the ratio of two correlated normals, (ii) supplies both an analytic error bound and accompanying numerical verification for the ratio approximation, and (iii) documents the convergence of the inner root-finding routine for h(k) over the admissible interval of k via additional tables or figures. These additions will directly support the reported superiority of the adaptive design. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper defines a bilevel optimization that uses an external finite-state Markov-chain model (with ratio approximation) to evaluate TARL0 and TARL1 as performance metrics. The inner root-finding enforces the target TARL0 constraint and the outer search minimizes TARL1; reported improvements are relative to benchmark charts under the same model. No step reduces a claimed prediction or result to a quantity defined by the fitted parameters themselves, nor relies on load-bearing self-citation or imported uniqueness. This is the standard non-circular workflow for control-chart design.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard bivariate normality of (X, Y), constant correlation, and the accuracy of the ratio approximation inside the Markov chain; no new entities are postulated and the optimized k and h are decision variables rather than fitted constants.

axioms (2)
  • domain assumption X and Y are jointly normal with constant correlation coefficient.
    Invoked throughout the Markov-chain construction for the ratio Z = X/Y.
  • domain assumption The finite-state Markov-chain model with ratio approximation yields accurate TARL values.
    Central to both the inner root-finding and outer line search steps.

pith-pipeline@v0.9.1-grok · 5905 in / 1439 out tokens · 21560 ms · 2026-06-27T23:28:59.806233+00:00 · methodology

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