Extended Exponentially Weighted Moving Average (Extended EWMA or EEWMA) control chart is one of the control charts which can quickly detect a small shift. The average run length (ARL) measures the performance of control chart. Due to the derivation of the explicit formulas for ARL on the EEWMA control chart for the autoregressive AR(p) process has not previously been reported. The aim of the article is to derive explicit formulas of ARL using a Fredholm integral equation of the second kind on EEWMA control chart for Autoregressive process, as AR(2) and AR(3) processes with exponential white noise. The accuracy of the solution obtained with the EEWMA control chart was compared to the numerical integral equation (NIE) method and extended to compare performance with Cumulative Sum (CUSUM) and Exponentially Weighted Moving Average (EWMA) control charts. The results show that the ARL obtained by the explicit formula and the NIE method are hardly different but ARL of explicit formula is less the computational (CPU) time than ARL of NIE method. The performance of EEWMA control chart is better than the CUSUM and EWMA control charts for all situations except when the large shift sizes the EEWMA control chart performed as well as the EWMA control chart for AR(2) and AR(3) processes. And then, the EEWMA control chart is also extended to compare efficiency of EEW-MA control chart with various λ. An exponential smoothing parameter of 0.05 is recommended. In addition, the simulation study, and efficacy illustration with real data on new COVID-19 cases in Thailand and Vietnam provided similar results.

The extended exponentially weighted moving average (EEWMA) control chart is an instrument for detection. It can quickly identify small shifts in the process. The benchmark for the control chart's performance is the average run length (ARL). In this paper, we present the efficiency of the EEWMA control chart to detect tiny shifts when the observations are autocorrelated with exponential residuals through the explicit formulas of the ARL. The accuracy of the solution was verified with the numerical integral equation (NIE) method. After that, the ARL effectiveness of the ARL on the EEWMA control chart was expanded to compare with the traditional EWMA control chart. Finally, using two real datasets that indicate the percentages of internet users using Windows 7 and iOS, the applicability of the offered method is shown. Our findings support the notion that the EEWMA control chart performs better when using autocorrelated data to track tiny changes.

The Extended Exponentially Weighted Moving Average (extended EWMA) control chart is one of the control charts and can be used to quickly detect a small shift. The performance of control charts can be evaluated with the average run length (ARL). Due to the deriving explicit formulas for the ARL on a two-sided extended EWMA control chart for trend autoregressive or trend AR(p) model has not been reported previously. The aim of this study is to derive the explicit formulas for the ARL on a two-sided extended EWMA control chart for the trend AR(p) model as well as the trend AR(1) and trend AR(2) models with exponential white noise. The analytical solution accuracy was obtained with the extended EWMA control chart and was compared to the numerical integral equation (NIE) method. The results show that the ARL obtained by the explicit formula and the NIE method is hardly different, but the explicit formula can help decrease the computational (CPU) time. Furthermore, this is also expanded to comparative performance with the Exponentially Weighted Moving Average (EWMA) control chart. The performance of the extended EWMA control chart is better than the EWMA control chart for all situations, both the trend AR (1) and trend AR(2) models. Finally, the analytical solution of ARL is applied to real-world data in the health field, such as COVID-19 data in the United Kingdom and Sweden, to demonstrate the efficacy of the proposed method.

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