Conditional analysis of Phase II exponential chart for monitoring times to an event

ABSTRACT

It is now well known that the error in estimating parameter(s) adversely affects the performance of control charts. The charts with estimated control limits are usually assessed by some of characteristics of unconditional run length (URL) distribution. However, it is now under criticism because the unconditional analysis based on URL does not give the full impression of the chart’s performance conditioning on a given Phase I sample. In recent years, the focus has been shifted from unconditional performance analysis to conditional analysis which is carried out in terms of conditional run length (CRL) and its associated characteristics and provides deep understanding of conditional performance of the chart given a Phase I sample. In this article, Phase II exponential charts are considered to monitor times to an event and their performance are examined conditional on given Phase I sample by considering several indicators based on conditional average run length (CARL) distribution. In this respect, the exact expressions of the distribution functions of $CARL$ for the estimated exponential control charts are derived and several statistical constants are computed which could be helpful in evaluating and designing the chart.

KEYWORDS:

• Conditional average run length
• conditional performance analysis
• distribution function
• lower prediction bound
• Phase II exponential chart
• times 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 Science and Engineering Research Board (SERB), Government of India (grant number-EMR/2017/002281).

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by the Science and Engineering Research Board (IN) [EMR/2017/002281].

Notes on contributors

Nirpeksh Kumar

Nirpeksh Kumar is an assistant professor at the Department of Statistics, Banaras Hindu University, Varanasi (BHU), India. He received his Master's and PhD degrees in Statistics from the University of Allahabad, Allahabad, 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 outlier detection, statistical process/quality control, and time series analysis.