The adaptive EWMA median chart for known and estimated parameters

ABSTRACT

The Adaptive Exponentially Weighted Moving Average (AEWMA) chart is known to combine the Shewhart and the classical EWMA schemes in a smooth way. Its performance has been investigated under the assumptions that the data are free from outliers and the chart's parameters are known or can be accurately estimated from in-control historical samples. However, there are many situations where the process parameters (a) are estimated from a very limited number of samples (b) they can potentially contain unexpected outliers. Therefore, in this article, we develop an AEWMA median chart with known and estimated parameters to monitor the mean value of a normal process. Taking the ‘Phase I between-practitioners variability’ into account, both the Average of $\mathrm{ARL}$ ($\mathrm{AARL}$) and the Standard Deviation of the $\mathrm{ARL}$ ($\mathrm{SDARL}$) are used to evaluate the conditional effect of the number of Phase I samples on the Phase II performance. Using a Markov Chain approach, it is shown that (a) the performance of the proposed AEWMA median chart is seriously affected when parameters are estimated compared with the known-parameter case, and (b) it requires a large amount of Phase I data to reduce the variation in the in-control $\mathrm{ARL}$ distribution up to a reasonable level. Therefore, a bootstrap-based design approach is applied here and, the performance of the AEWMA median chart is compared with that of the existing Shewhart and EWMA median charts. The comparative results show that the AEWMA median chart represents a good alternative to achieve a reasonable balance for various shift sizes.

KEYWORDS:

• Adaptive EWMA median chart
• estimated parameters
• standard deviation of average run length
• bootstrap method

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 62-XX

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Philippe Castagliola  http://orcid.org/0000-0002-9532-4029

Additional information

Funding

This work was supported by National Natural Science Foundation of China [71802110] and Natural Science Foundation of Jiangsu Province [BK20170894] and Humanity and Social Science Youth foundation of Ministry of Education of China [17YJC630043] and China Scholarship Council [201706840050] and Social Sciences Foundation of Nanjing University of Posts and Telecommunications [NYY217007 and NY218041].