Statistical design of phase II exponential chart with estimated parameters under the unconditional and conditional perspectives using exact distribution of median run length

ABSTRACT

In this paper, we consider the $q$-quantiles of the conditional run length (CRL) distribution to design and evaluate the phase II exponential chart instead of the widely used metric, the average of CRL. The unconditional and conditional perspectives of the performance evaluation are considered to adjust the control limits of the phase II exponential chart based on the in-control (IC) $q$-quantiles of CRL (denoted by $\mathrm{C}\mathrm{Q}\mathrm{R}{\mathrm{L}}_q$) distribution. Under the unconditional approach, the mean of the IC $\mathrm{C}\mathrm{Q}\mathrm{R}{\mathrm{L}}_q$ is set equal to some pre-specified value, say $\mathrm{Q}\mathrm{R}{\mathrm{L}}_0$ of the $q$-quantile of run length (QRL) whereas under the conditional perspective, the control limits are adjusted so that that the IC $\mathrm{C}\mathrm{Q}\mathrm{R}{\mathrm{L}}_q$ meets or exceeds the nominal $\mathrm{Q}\mathrm{R}{\mathrm{L}}_0$ value with a high probability. The charts under both perspectives are designed so that they are QRL-unbiased for a nominal $\mathrm{Q}\mathrm{R}{\mathrm{L}}_0$ value. The IC and out-of-control (OOC) performance studies of the proposed charts are carried out based on the most promising quantile i.e. median run length (MRL). The study shows that the MRL-unbiased exponential chart under the conditional perspective has a better IC performance.

KEYWORDS:

• Control chart
• high-yield processes
• in-control and out-of-control performances
• quantiles of run length
• time between events

Acknowledgments

The author would like to thank two anonymous reviewers and the Editor for their helpful and constructive comments that have improved the article. The present work was supported by the Science and Engineering Research Board (SERB), Government of India (grant number‐EMR/2017/002281). Partial support was also provided by the Banaras Hindu University, India under the IoE Scheme (Grant Number 6031).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Banaras Hindu University [IoE 6031]; Science and Engineering Research Board [EMR/2017/002281].

Notes on contributors

Nirpeksh Kumar

Nirpeksh Kumar is an Associate Professor at the Department of Statistics, Banaras Hindu University, Varanasi, India. He received his Master’s and PhD degrees in Statistics from the University of Allahabad, Prayagraj, India. He was awarded SARChI Post-doctoral fellowship at the Department of Statistics, University of Pretoria, South Africa. He has published in numerous accredited peer-reviewed journals and has presented his research at several national and international conferences. His research interests include statistical process/quality control, and time series analysis.