Nonparametric control limits incorporating exceedance probability criterion for statistical process monitoring with commonly employed small to moderate sample sizes

Abstract

This article aims to enhance the effectiveness of Phase I in statistical process monitoring by integrating the assessment of estimation uncertainty into control charts using the exceedance probability criterion. This criterion guarantees the desired in-control performance that a practitioner will achieve with a predefined high nominal coverage probability, which can help prevent high false alarm rates from occurring. In pursuit of this objective, we introduce two nonparametric approaches: one based on an analytical method and the other on a bootstrapping technique. Both approaches exhibit superior performance compared to the existing nonparametric method, particularly for Phase I, where small to moderate sample sizes are common. These proposed methodologies are especially advantageous for practitioners in real-world production environments.

Keywords:

• bootstrap
• exceedance probability criterion
• nonparametric control chart
• phase i stage
• statistical process monitoring
• univariate shewhart control chart

Notes

1 Using the open dataset accessible on Kaggle’s website https://www.kaggle.com/datasets/paresh2047/uci-semcom?resource=download.

2 The equal-tail control limits are represented by estimating probability limits through the Upper Control Limit (UCL) and the Lower Control Limit (LCL). These limits correspond to the upper $1-{\alpha }_0/2$ and lower ${\alpha }_0/2$ percentiles, respectively, of an unidentified probability distribution (for further details, see Section 2). In this study, the false alarm rate α₀ is set to 0.0027, aligning with the Shewhart control chart monitoring the mean of a variable based on 3-sigma limits. When in-control parameters are known, this control chart yields an α₀ of 0.0027 for normally distributed data.

Additional information

Funding

The research was supported by a grant from the National Science and Technology Council in Taiwan with NSTC 111-2118-M-006-002 and MOST 111-2118-M-006-002- MY2.

Notes on contributors

Hong-Ji Yang

Hong-Ji Yang is presently engaged in doctoral studies within the Department of Statistics at the University of Cheng Kung in Taiwan. His academic path is preceded by a notable career as a statistician, where he significantly impacted the Quality and Reliability Division at Taiwan Semiconductor Manufacturing Company Limited (TSMC) during a dedicated service spanning more than two decades from 2000 to 2021. His research expertise encompasses statistical process monitoring, quality engineering, measurement system analysis, and the dynamic field of data science.

Chung-I Li

Chung-I Li is an associate professor in the Department of Statistics at National Cheng Kung University in Taiwan. His current research interest is in the fields of statistical quality control and applied statistics.