# Optimal Design of One-Sided Exponential Adaptive EWMA Scheme Based on Median Run Length
## 6 Two Illustrative Examples
### Example 2  
The dataset introduced by [PERSON] includes the time intervals between coal mine explosions occurring from 15 March 1851, to 22 March 1962, measured in days [38]. Assuming an exponential distribution with unknown parameters, the bootstrap method was used to estimate the process mean using the first 160 data points (see Table 7). The remaining 30 data points (see Table 8) were reserved for testing purposes. To obtain a robust estimate of the process mean, the initial 160 data points were employed to compute the MLE, yielding \(\widehat{\theta}_{0}=182.3\). Subsequently, the bootstrap method generated multiple resamples from these data points. For each resample, the MLE of \(\widehat{\theta}_{0}\) was recalculated, thereby forming a distribution of estimates. Based on this distribution, the 95% confidence interval for \(\widehat{\theta}_{0}\) was determined as \([\widehat{\theta}_{0L},\widehat{\theta}_{0L}]=[146,223.5]\). To effectively monitor and promptly detect upward shifts in the process mean, an optimally designed upper-sided AEWMA scheme is implemented, utilizing MRL as the performance metric. This approach enhances the sensitivity of the monitoring scheme in identifying potential increases in event frequency, thereby facilitating timely intervention and improving safety monitoring in critical environments. The current monitoring objective focuses on detecting shifts in the process mean within the range \([c_{1},c_{2}]=[1.02,5]\), with an in-control MRL (\(MRL_{0}\)) of 200. Following the optimization procedure detailed in Section 4, the optimal parameters for the upper-sided monitoring scheme are determined to be \((\lambda^{+},k,LCL^{+})=(0.052,1.432,4.975)\). For practical implementation, these parameters are scaled by \(\widehat{\theta}_{0}\), yielding the adjusted parameters \(B^{+^{\prime}}\), \(k^{\prime}\), and \(UCL^{\prime}\). Specifically, the adjusted parameters are computed as:  
\[B^{+^{\prime}} =0.5\quad\times 182.3=91.2,\] \[k^{\prime} =1.432\times 182.3=261.054,\] \[UCL^{\prime} =4.975\times 182.3=906.943.\]  
The optimal upper-sided AEWMA scheme statistic \(A_{l}^{+}\) is calculated using Equation (7) (refer to Table 8 for details). To refine the monitoring scheme and account for the uncertainty in the estimation of the process mean, we consider the upper-bound of the bootstrap confidence interval, \(\widehat{\theta}_{0L}=223.5\), to recalculate the adjusted parameters \(B_{U}^{+^{\prime}}\), \(K_{U}^{\prime}\), and \(UCL^{\prime}_{U}\). These adjusted parameters provide a more conservative estimate of the control limits, ensuring the monitoring scheme's effectiveness even under conditions where the true process mean might be higher than initially estimated. The recalculated parameters are as follows:  
Figure 7: Lower-sided exponential Shewhart scheme in the OLED production process. \[{B^{+}}^{\prime}_{UI} =0.5\quad\times 223.5 =111.8,\] \[k^{\prime}_{UI} =1.432\times 223.5 =320.052,\] \[UCL^{\prime}_{UI} =4.975\times 223.5 =1111.913.\]  
The AEWMA statistic \(A^{+}_{tUI}\) is then calculated using Equation (7), with the updated parameters (refer to Table 8 for details). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(t\) & \(\mathbf{Z_{t}}\) & \(\mathbf{t}\) & \(\mathbf{Z_{t}}\) & \(\mathbf{t}\) & \(\mathbf{Z_{t}}\) & \(\mathbf{t}\) & \(\mathbf{Z_{t}}\) & \(\mathbf{t}\) & \(\mathbf{Z_{t}}\) \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 8: Phase II samples \(X_{t}\) (in days) and corresponding monitoring statistics \(A^{+}_{t}\), \(A^{+}_{tUI}\) and \(Q^{+}_{t}\) of the optimal upper-sided exponential schemes for the explosions in mines. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(t\) & \(\mathbf{X_{t}}\) & \(A^{+}_{t}\) & \(A^{+}_{tUI}\) & \(\mathbf{Q^{+}_{t}}\) & \(\mathbf{t}\) & \(\mathbf{X_{t}}\) & \(\mathbf{A^{+}_{t}}\) & \(\mathbf{A^{+}_{tUI}}\) & \(\mathbf{Q^{+}_{t}}\) \\ \hline
1 & 275 & 187.