[Paper Review] Risk-Calibrated Process Capability Approval with Finite Samples

Fei Jiang, Lei Yang|arXiv (Cornell University)|Mar 15, 2026

Advanced Statistical Process Monitoring0 citations

TL;DR

The paper develops a risk-calibrated binary decision framework for process capability approval under finite samples, deriving a margin-based rule that accounts for estimation uncertainty and asymmetric loss.

ABSTRACT

Process capability indices such as $C_{pk}$ are widely used in manufacturing to support supplier qualification, pilot-build release, and production approval. In practice, approval decisions are often based on deterministic threshold rules of the form $\widehat{C}_{pk} \ge C_0$. Because $\widehat{C}_{pk}$ is estimated from finite samples, however, such decisions are inherently stochastic, especially when the true capability lies near the approval threshold. This paper develops a risk-calibrated decision framework for process capability approval that explicitly accounts for estimation uncertainty and asymmetric operational loss. Capability approval is formulated as a binary statistical decision problem, leading to a rule of the form $\widehat{C}_{pk} \ge C_0 + k\,SE(\widehat{C}_{pk})$, where the calibration constant $k$ is determined either by a tolerable failure probability or by a false-accept/false-reject cost ratio. The resulting formulation unifies several commonly used procedures, including deterministic thresholding, lower confidence bound rules, and probability-based approval rules, and naturally extends them to cost-sensitive decision rules derived from asymmetric operational loss. Simulation experiments and an industrial case study show that risk calibration primarily affects near-threshold decisions, improves approval stability, and can substantially reduce expected operational loss when false acceptance is more costly than false rejection.

Motivation & Objective

• Motivate capability approval as a binary decision under finite-sample uncertainty.
• Incorporate asymmetric operational losses into capability approval decisions.
• Derive a unified margin-based decision rule that generalizes deterministic, LCB, and probability-based rules.
• Link calibration constant to operating characteristics and misclassification risk.
• Demonstrate practical implications through simulation and an industrial case study.

Proposed method

• Model capability approval as a binary decision problem with asymmetric loss.
• Approximate P(C_true < C0 | D) via a normal approximation to obtain a margin rule.
• Propose the unified margin rule: C_hat_pk >= C0 + k SE(C_hat_pk) with k interpreted under different schemes.
• Show equivalence to deterministic thresholding, LCB, and probability-based rules as special cases.
• Derive a cost-sensitive rule using the loss ratio lambda and alpha = 1/(1+lambda).
• Provide a practical calibration guidance linking lambda, alpha, and k.

Experimental results

Research questions

• RQ1 How should capability approval be designed when estimation uncertainty is explicit and near the threshold?
• RQ2 How do different approval rules (deterministic, LCB, probability-based, cost-sensitive) relate within a unified framework?
• RQ3 What is the impact of asymmetric costs on false acceptance/false rejection and operational loss?
• RQ4 How does the proposed risk-calibrated rule perform in finite samples and in industrial data?

Key findings

• A unified margin-based approval rule is derived: accept if C_hat_pk >= C0 + k SE(C_hat_pk).
• The calibration constant k corresponds to different schemes: k = 0 for deterministic, k = -z_alpha for probability-based, and k = -z_{1/(1+lambda)} for cost-sensitive rules.
• Asymmetric costs (higher c_FA) shift the approval boundary to be more conservative, reducing false acceptance at the expense of more false rejection.
• Near-threshold decisions are most affected by risk calibration; the rules stabilize approvals and can reduce expected operational loss.
• Simulation and case study show risk calibration can substantially reduce loss when false acceptance is costly.