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Finite-Sample Decision Instability in Threshold-Based Process Capability Approval

Fei Jiang, Lei Yang

Abstract

Process capability indices such as $C_{pk}$ are widely used in manufacturing quality control to support supplier qualification and product release decisions based on fixed acceptance thresholds (e.g., $C_{pk} \geq 1.33$). In practice, these decisions rely on sample-based estimates computed from moderate sample sizes ($n \approx$ 20-50), yet the stochastic nature of the estimator is often overlooked when interpreting threshold compliance. This study establishes a local asymptotic characterization of decision behavior when the true process capability lies near a fixed threshold. Under standard regularity conditions, if the true capability equals the threshold, the acceptance probability converges to 0.5 as sample size increases, implying that a fixed $C_{pk}$ gate embeds an inherent boundary decision risk even under ideal distributional assumptions. When the true capability deviates from the threshold by $O(n^{-1/2})$, the decision probability converges to a non-degenerate limit governed by a scaled signal-to-noise ratio. Monte Carlo simulations and an empirical study on 880 manufacturing dimensions demonstrate substantial resampling-based decision instability near the commonly used 1.33 criterion. These findings provide a probabilistic interpretation of threshold-based capability decisions and quantitative guidance for assessing boundary-induced release risk in engineering practice.

Finite-Sample Decision Instability in Threshold-Based Process Capability Approval

Abstract

Process capability indices such as are widely used in manufacturing quality control to support supplier qualification and product release decisions based on fixed acceptance thresholds (e.g., ). In practice, these decisions rely on sample-based estimates computed from moderate sample sizes ( 20-50), yet the stochastic nature of the estimator is often overlooked when interpreting threshold compliance. This study establishes a local asymptotic characterization of decision behavior when the true process capability lies near a fixed threshold. Under standard regularity conditions, if the true capability equals the threshold, the acceptance probability converges to 0.5 as sample size increases, implying that a fixed gate embeds an inherent boundary decision risk even under ideal distributional assumptions. When the true capability deviates from the threshold by , the decision probability converges to a non-degenerate limit governed by a scaled signal-to-noise ratio. Monte Carlo simulations and an empirical study on 880 manufacturing dimensions demonstrate substantial resampling-based decision instability near the commonly used 1.33 criterion. These findings provide a probabilistic interpretation of threshold-based capability decisions and quantitative guidance for assessing boundary-induced release risk in engineering practice.
Paper Structure (37 sections, 2 theorems, 48 equations, 6 figures, 1 table)

This paper contains 37 sections, 2 theorems, 48 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Sampling Framework
  3. Sampling Variability Under Normal Assumptions
  4. Decision-Theoretic Formulation of Capability Classification
  5. Misclassification Formulation
  6. Type I Misclassification (False Accept)
  7. Type II Misclassification (False Reject)
  8. Finite-Sample Decision Performance
  9. Boundary Instability Theorem
  10. Assumption A1 (Unique Active Constraint).
  11. Assumption A2 (Regularity).
  12. Visualization remark.
  13. Proof sketch.
  14. Closed-Form Approximation of $\sigma_C$ under Normality
  15. Step 1: Asymptotic covariance of $(\bar{X},S)$.
  16. ...and 22 more sections

Key Result

Theorem 1

Let $C_0$ be a fixed threshold. Under Assumptions A1-A2, where $\Phi(\cdot)$ is the standard normal distribution function. Moreover, consider a local parameterization where $h=\sqrt{n}(C_{pk}^{true}-C_0)$ is the $\sqrt{n}$-scaled distance of the true capability level from the approval threshold. Then In particular, at the exact boundary $C_{pk}^{true} = C_0$,

Figures (6)

  • Figure 1: Finite-sample misclassification risk surface under normal sampling with bilateral specifications and threshold $C_0 = 1.33$. The horizontal axis denotes $C_{pk}^{true}$ and the vertical axis denotes sample size $n$. For $C_{pk}^{true}<C_0$, the plotted quantity is $\Pr(\widehat{C}_{pk}\ge C_0)$ (Type I false accept); for $C_{pk}^{true}\ge C_0$, it is $\Pr(\widehat{C}_{pk}<C_0)$ (Type II false reject). A pronounced ridge of elevated misclassification probability appears near $C_{pk}^{true}=C_0$.
  • Figure 2: Sampling distributions of $\widehat{C}_{pk}$ near the threshold $C_0=1.33$. The dashed vertical line denotes the approval threshold. The shaded region represents the Type II misclassification event $\{\widehat{C}_{pk} < C_0\}$ for a true-capable process ($C_{pk}^{true}>C_0$). Two sample sizes are shown to illustrate variance contraction with increasing $n$. The legend reports the estimated misclassification probability.
  • Figure 3: Validation of $\sqrt{n}$ scaling near the capability decision boundary. Panel (a) shows the acceptance probability under the rescaled variable $z=\sqrt{n}(C_{pk}^{true}-C_0)/\sigma_C$. Monte Carlo curves for different sample sizes collapse and closely follow the theoretical prediction $\Phi(z)$, confirming that near the decision boundary classification behavior is governed by a single signal-to-noise parameter. The horizontal line at $0.5$ highlights the boundary instability at $C_{pk}^{true}=C_0$. Panel (b) displays the residual $\Delta(z)=P_{MC}(z)-\Phi(z)$. Deviations are small and decrease with sample size, demonstrating good finite-sample accuracy of the local asymptotic approximation.
  • Figure 4: Acceptance probability surfaces under three capability approval rules as functions of the true population capability $C_{pk}^{true}$ and sample size $n$, with threshold $C_0 = 1.33$. Colors represent the probability of accepting the process under repeated sampling. (a) Deterministic rule: acceptance occurs when $\widehat{C}_{pk} \ge C_0$. The dashed contour denotes the $0.5$ acceptance boundary and the solid contour denotes the $0.95$ boundary. (b) Lower confidence bound (LCB) rule: acceptance requires $\mathrm{LCB}_{0.95} \ge C_0$. (c) Probability rule: acceptance requires $\Pr(\widehat{C}_{pk} \ge C_0) \ge 0.95$, estimated via Monte Carlo simulation.
  • Figure 5: Empirical decision instability as a function of deterministic distance from the approval threshold. The solid curve represents the bin-averaged flip rate (using quantile-based bins), and the shaded band denotes the interquartile range within each bin. Bootstrap replication size: $B=5000$ per dimension.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Theorem 1: Local Boundary Instability
  • Proposition 1: Explicit $1/\sqrt{n}$ width of the instability region