# Optimal Design of One-Sided Exponential Adaptive EWMA Scheme Based on Median Run Length
## 6 Two Illustrative Examples
### Example 2
As shown in the table, at \(t=22\), \(A_{t}^{+}\) reaches a value of \(1382.521\), and \(A_{tU}^{+}\) reaches a value of \(1326.591\), both exceeding their respective control limit \(UCL^{\prime}\) and \(UCL_{U}^{\prime}\), as highlighted in bold. Additionally, it is noteworthy that at \(t=27\), the AEWMA statistic \(A_{t}^{+}\) exceeds its control limit \(UCL^{\prime}\), signaling a potential out-of-control condition. In contrast, the adjusted AEWMA statistic \(A_{tU}^{+}\) remains below its respective control limit \(UCL_{U}^{\prime}\) at this time point. This discrepancy highlights the sensitivity of the AEWMA scheme to upward shifts in the process mean, while the adjusted scheme, which incorporates the upper bound of the bootstrap confidence interval, provides a more conservative estimate. This observation underscores the importance of considering parameter uncertainty in the design of monitoring schemes to ensure robust performance under varying conditions. To promote a comparative analysis in practical applications, we also evaluate the setting where \(c=1.02\) and \(MRL_{0}=200\). For the EWMA-type scheme, the optimal parameters derived from Table 3, which are \((\lambda^{+},UCL^{+})=(0.052,1.369)\). These parameters were then scaled by the factor \(\widehat{\theta}_{0}=182.3\), resulting in \({B^{+}}^{\prime}=91.2\) and \(UCL^{\prime}=249.569\). The corresponding optimal statistic for the EWMA scheme, \(Q_{t}^{+}\), can be computed using Equation (5) (see Table 8). For the Shewhart-type scheme, the control limit was set at \(5.6616\) to satisfy \(MRL_{0}=200\). After scaling this value by \(\widehat{\theta}_{0}=182.3\), the adjusted upper control limit \(UCL^{\prime}\) is calculated as \(UCL^{\prime}=5.6616\times 182.3=1032.110\). According to the data presented in Table 8, at time points \(t=22,27,28\), when \(X_{t}\) undergoes a sudden increase, \(A_{t}^{+}\) shows a more significant increment compared to \(Q_{t}^{+}\). This observation indicates that, under such conditions, the monitoring performance of the AEWMA scheme aligns more closely with that of the Shewhart scheme. Figures 8-11 visually illustrates these monitoring schemes, clearly showing that at \(t=22\), the schemes emit an out-of-control signal, suggesting a notable shift in the process mean. Furthermore, it can be observed that when \(X_{t}\) (at \(t=23\)) experiences a sudden decrease, the EWMA-type scheme continues to emit an out-of-control signal. In contrast, the AEWMA and Shewhart schemes quickly adapt to the new data, returning to the in-control region and thereby reducing unnecessary alarms. This behavior underscores the importance of selecting a monitoring scheme that balances sensitivity to shifts with the ability to avoid false alarms, making the AEWMA scheme particularly suitable for scenarios requiring rapid detection of changes while minimizing false alerts. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \(t\) & \(X_{t}\) & \(A_{t}^{+}\) & \(A_{tU}^{+}\) & \(Q_{t}^{+}\) & \(t\) & \(X_{t}\) & \(A_{t}^{+}\) & \(A_{tU}^{+}\) & \(Q_{t}^{+}\) \\ \hline
5 & 4 & 169.704 & 201.250 & 169.704 & 20 & 47 & 234.942 & 203.937 & 189.777 \\
6 & 368 & 180.016 & 209.921 & 180.016 & 21 & 129 & 229.433 & 200.040 & 186.616 \\
7 & 307 & 186.619 & 214.969 & 186.619 & 22 & 1630 & 1382.521 & 1326.591 & 261.672 \\
8 & 336 & 194.387 & 221.263 & 194.387 & 23 & 29 & 276.479 & 332.409 & 249.573 \\
9 & 19 & 185.266 & 210.745 & 185.266 & 24 & 217 & 273.386 & 326.408 & 247.879 \\
10 & 329 & 192.741 & 216.894 & 192.741 & 25 & 7 & 254.479 & 309.799 & 235.354 \\
11 & 330 & 199.878 & 222.776 & 199.878 & 26 & 18 & 242.182 & 294.625 & 224.051 \\
12 & 312 & 205.708 & 227.415 & 205.708 & 27 & 1358 & 1110.521 & 1054.591 & 283.017 \\
13 & 536 & 288.521 & 243.462 & 222.884 & 28 & 2366 & 2118.521 & 2062.591 & 391.332 \\
14 & 145 & 281.058 & 238.342 & 218.834 & 29 & 952 & 1199.479 & 1255.409 & 420.486 \\
15 & 75 & 270.343 & 229.848 & 211.354 & 30 & 632 & 879.479 & 935.409 & 431.485 \\ \hline \end{tabular}
\end{table}