Synthetic exponential control charts with unknown parameter

ABSTRACT

The existing synthetic exponential control charts are based on the assumption of known in-control parameter. However, the in-control parameter has to be estimated from a Phase I dataset. In this article, we use the exact probability distribution, especially the percentiles, mean, and standard deviation of the conditional average run length (ARL) to evaluate the effect of parameter estimation on the performance of the Phase II synthetic exponential charts. This approach accounts for the variability in the conditional ARL values of the synthetic chart obtained by different practitioners. Since parameter estimation results in more false alarms than expected, we develop an exact method to design the adjusted synthetic charts with desired conditional in-control performance. Results of known and unknown in-control parameter cases show that the control limit of the conforming run length sub-chart of the synthetic chart should be as small as possible.

KEYWORDS:

• Conditional average run length
• Exponential distribution
• Parameter estimation
• Standard deviation of average run length
• Synthetic chart

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 62-XX
• Secondary 62P30

Acknowledgments

The authors thank the editor, the associate editor, and the reviewers for their detailed comments and suggestions, which considerably helped improve the manuscript.

Funding

The work of Sun is sponsored by the National Social Science Foundation of China (Grant No. 14BTJ022). The work of Wang is supported by the National Natural Science Foundation of China (Grant Nos. 11371322, 11671303), Zhejiang Provincial Natural Science Foundation of China (Grant No. LY16G020003), Foundation of Ministry of Education of China (Grant No. 13YJC910010), and First Class Discipline of Zhejiang–A (Zhejiang Gongshang University–Statistics). The work of Guo is sponsored by the National Social Science Foundation of China (Grant No. 14BTJ030). The work of Xie is supported by the National Natural Science Foundation of China (Grant No. 71371163) and a grant from University Grants Council of Hong Kong (GRF 9042327).