# An Enhanced Auxiliary Information-Based EWMA-t Chart for Monitoring the Process Mean
## 4 Performance Measurement and Comparison
### Performance Comparison  
The out-of-control \(ARL\), denoted by \(ARL_{1}\), is expected to be sufficiently small to detect shifts early when the process is out of control. It is customary to set an \(ARL_{0}\) value and then compute the \(ARL_{1}\) values. The smaller the \(ARL_{1}\) value, the better is the statistical performance. To compare the performance of the AIB-GWMA-t charts, this study keeps \(ARL_{0}\) close to 500 and the \(ARL_{1}\)s are evaluated for specific process mean shifts. To ensure the discussion remains brief, design parameter \(q=\{0.5,\,0.7,\,0.9,\,0.95\}\), adjustment parameter \(\alpha=\{0.5,\,0.7,\,0.9,\,1.0\}\), correlation coefficient \(\rho=\{0.00,\,0.25,\,0.50,\,0.75,\,0.95\}\), and process mean shifts \(\delta=\{0.1,\,0.2,\,0.4,\,0.6,\,1.0,\,2.0\}\) are used to compute the \(ARL\) values of the AIB-GWMA-t charts with \(n=5\) and \(n=10\) (see Tables 2 and 3, respectively). In particular, \(\alpha=1.0\) and \(q=1-\lambda\) reduce to the AIB-EWMA-t chart proposed by [PERSON] et al. (2018). Furthermore, when \(\rho=0\), the AIB-GWMA-t and AIB-EWMA-t charts reduce to the GWMA-t and EWMA-t charts, respectively. The boldface values in Tables 2 and 3 indicate smaller \(ARL_{1}\) values than those of the AIB-EWMA-t and EWMA-t charts under various process mean shifts with the specific design parameter \(q\) and correlation coefficient \(\rho\). The main findings from Tables 2 and 3 are as follows:  
1. For fixed \(q\) and \(\alpha\), the values of \(L_{ag}\) are close and unaffected by the choices of \(\rho\), a finding consistent with that of [PERSON] and [PERSON] (2018) and [PERSON] et al. (2018). 2. A larger sample size \(n\) results in a smaller \(ARL_{1}\) value at fixed parameter combinations of \((q,\alpha,\rho)\). 3. For fixed \(q\alpha\), and \(\rho\), the \(ARL_{1}\) values decrease as the value of \(\delta\) increases. Moreover, for fixed \(\alpha\), \(\delta\), and \(\rho\), the \(ARL_{1}\) values tend to decrease as the value of \(q\) increases. Similarly, the \(ARL_{1}\) value is a decreasing function of \(\rho\) when \(q\), \(\alpha\), and \(\delta\) are fixed. 4. The AIB-GWMA-t and AIB-EWMA-t charts uniformly perform better than the GWMA-t and EWMA-t charts, respectively. This result reveals that the use of auxiliary information enhances the performance of the GWMA-t and EWMA-t charts, especially for large values of \(\rho\). 5. To detect small process mean shifts, the AIB-GWMA-t chart with large \(q\) and \(\alpha\) performs better than the AIB-EWMA-t chart with \(0.25\leq\rho\leq 0.75\). However, the AIB-GWMA-t chart performs comparably to the AIB-GWMA-t chart at \(\rho=0.95\). Facing an auxiliary variable with different correlations, suitable parameter combinations of \((q,\alpha)\) facilitate the use of our proposed charts to monitor small process mean shifts. Figure 2 depicts the \(ARL_{1}\) curves of the AIB-GWMA-t charts at different correlation coefficients with specific parameter combinations. It shows that when we apply the proposed GWMA-t or AIB-GWMA-t charts to monitor small process mean shifts \((\delta=0.1)\), large values of \(q\) and \(\alpha\) under \(\rho\leq 0.75\) are recommended for practical applications. Moreover, the performance of the AIB-GWMA-t chart is comparable to the AIB-EWMA-t chart at \(q=0.95\) regardless of the \(\alpha\) value for \(\rho=0.95\). ==ERROR: No output for this page==