# Exact Solution of Average Run Length on Extended EWMA Control Chart for the First-Order Autoregressive Process

## Authors

• Kotchaporn Karoon Department of Applied Statistics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, Thailand
• Yupaporn Areepong Department of Applied Statistics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, Thailand.
• Saowanit Sukparungsee Department of Applied Statistics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, Thailand

## Keywords:

EWMA control chart, change point detection, numerical integral equation, explicit formulas, autoregressive

## Abstract

Statistical process control methods are widely used in several fields for monitoring and detecting process problems. One of them is the control chart which is effective for monitoring process. Generally, the average run length (ARL) is measured the performance of control chart. The aim of the article is to derive explicit formulas of ARL using a Fredholm integral equation of the second kind on an extended exponentially weighted moving average (EEWMA) control chart for first-order autoregressive (AR(1)) process with exponential white noise. The accuracy of the solution obtained with the EEWMA control chart was compared to the numerical integral equation (NIE) method. The analytical results agree with NIE approximations with an absolute percentage difference less than $1.191\times&space;10^{-4}$ and the computational times of NIE approximately 2-4 seconds whereas the computational time of the explicit formulas is less than one second. In addition, a performance comparison of the ARL using explicit formulas on the EEWMA and EWMA control charts show that they performed better on the EEWMA control chart for all shift sizes and cases. Besides, an exponential smoothing parameter of 0.05 is recommended. Moreover, the ARL performances on the EEWMA and EWMA charts were compared using real data on the concentration of the 24-hour average of particulate matter or PM10 $(10\mu&space;g/m^{3})$ in the air pollutant in the air which is an important indicator of air pollution and a major environmental problem. The results indicate that the EEWMA chart performed better than the EWMA chart for all situations.

2022-03-28

## How to Cite

Karoon, K. ., Areepong, Y. ., & Sukparungsee, S. . (2022). Exact Solution of Average Run Length on Extended EWMA Control Chart for the First-Order Autoregressive Process. Thailand Statistician, 20(2), 395–411. Retrieved from https://ph02.tci-thaijo.org/index.php/thaistat/article/view/246348

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