Risk-Calibrated Process Capability Approval with Finite Samples
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909038223032320 |
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| author | Jiang, Fei Yang, Lei |
| author_facet | Jiang, Fei Yang, Lei |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_14479 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Risk-Calibrated Process Capability Approval with Finite Samples Jiang, Fei Yang, Lei Applications Methodology 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. |
| title | Risk-Calibrated Process Capability Approval with Finite Samples |
| topic | Applications Methodology |
| url | https://arxiv.org/abs/2603.14479 |