### Journals Information

Mathematics and Statistics Vol. 9(5), pp. 685 - 696
DOI: 10.13189/ms.2021.090508
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## Analytical Solutions of ARL for SAR(p)L Model on a Modified EWMA Chart

Piyatida Phanthuna 1, Yupaporn Areepong 2,*
1 Faculty of Science and Technology, Rajamangala University of Technology Phra Nakhon, 10800, Thailand
2 Faculty of Applied Science, King Mongkutâ€™s University of Technology North Bangkok, 10800, Thailand

ABSTRACT

A modified exponentially weighted moving average (EWMA) scheme expanded from an EWMA chart is an instrument for immediate detection on a small shifted size. The objective of this research is to suggest the average run length (ARL) with the explicit formula on a modified EWMA control chart for observations of a seasonal autoregressive model of order pth (SAR(p)L) with exponential residual. A numerical integral equation method is brought to approximate ARL for checking an accuracy of explicit formulas. The results of two methods show that their ARL solutions are close and the percentage of the absolute relative change (ARC) is obtained to less than 0.002. Furthermore, the modified EWMA chart with the SAR(p)L model is tested to shift detection when the parameters c and are changed. The ARL and the relative mean index (RMI) results are found to be better when c and are increased. In addition, the modified EWMA control chart is compared to performance with the EWMA scheme and such that their results encourage the modified EWMA chart for a small shift. Finally, this explicit formula can be applied to various real-world data. For example, two data about information and communication technology are used for the validation and the capability of our techniques.

KEYWORDS
Explicit Formula, Seasonal Autoregressive, Average Run Length

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Piyatida Phanthuna , Yupaporn Areepong , "Analytical Solutions of ARL for SAR(p)L Model on a Modified EWMA Chart," Mathematics and Statistics, Vol. 9, No. 5, pp. 685 - 696, 2021. DOI: 10.13189/ms.2021.090508.

(b). APA Format:
Piyatida Phanthuna , Yupaporn Areepong (2021). Analytical Solutions of ARL for SAR(p)L Model on a Modified EWMA Chart. Mathematics and Statistics, 9(5), 685 - 696. DOI: 10.13189/ms.2021.090508.