# An Enhanced Auxiliary Information-Based EWMA-t Chart for Monitoring the Process Mean
## 4 Performance Measurement and Comparison  
### In-Control ARL Profiles  
The \(ARL\) (Average Run Length) is a popular indicator for evaluating the performance of control charts. When the process operates in an in-control state, the in-control \(ARL\), termed \(ARL_{0}\), of a AIB-GWMA-t chart is a function of \((q,\alpha,L_{ag})\) and this is expected to be sufficiently large to avoid false alarms. To compute the \(ARL\) values, the Monte Carlo simulation was recommended by [PERSON] and [PERSON] and [PERSON] (2017) instead of the Markov chain or integral equation approaches since the GWMA statistic is complicated. An algorithm in R has been developed in Appendix A to calculate the \(ARL\) values, which are 50,000 run lengths on average. Without loss of generality, the random samples \((X_{ij},Y_{ij})\), \(j=1,2,\ldots,n\), for \(i=1,2,\ldots\), are drawn from a bivariate normal distribution, that is, \((X,Y)\sim N_{2}(\mu_{X},\mu_{Y},\sigma_{X}^{2},\sigma_{Y}^{2},\rho)\). For \(\mu_{X}=\mu_{Y}=0\), \(\sigma_{X}=\sigma_{Y}=1\), and the known correlation coefficient \(\rho\), we find that the underlying process is in-control. The control limit coefficient \(L_{ag}\) under various combinations of \(n\), \(q\), and \(\alpha\) can be adjusted to match the desired \(ARL_{0}\).  
Table 1 presents the \(L_{ag}\) values for the AIB-GWMA-t charts. The various parameter combinations of \(q\in\{0.1,0.2,\ldots,0.9\}\) and \(\alpha\in\{0.1,0.2,\ldots,1.0\}\), correlation coefficient value of \(\rho=0\), and sample size \(n\in\{5,\,10\}\) are considered to achieve an in-control \(ARL\) at approximately 500. Note that sample sizes \(n=5\) and 10 are selected for brief discussion to compare with that of [PERSON] et al. [PERSON] et al. (2018) and explore the effect of different sample sizes on \(L_{ag}\). When \(\rho=0\), the results in Table 1 correspond to the proposed GWMA-t charts.  
Table 1 shows that when \(q\) and \(\alpha\) are fixed, a large sample size corresponds to a small control limit coefficient and vice versa. The \(L_{ag}\) values decrease as design parameter \(q\) or adjustment parameter \(\alpha\) increases. In particular, the difference among the \(L_{ag}\) values is large for larger \(q\) or \(\alpha\) values, whereas the difference among the \(L_{ag}\) values become small for smaller \(q\) or \(\alpha\) values. For example, when \(n=5\), the values of \(L_{ag}\) for \(q=0.1\) and 0.9 at \(\alpha=0.9\) are 5.060 and 3.145, respectively; however, those for \(q=0.1\) and 0.9 at \(\alpha=0.1\) are 5.079 and 5.060, respectively. Moreover, when \(\alpha=1.0\), the \(L_{ag}\) value of the  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \multirow{2}{*}{\(n\)} & \multirow{2}{*}{\(q\)} & \multicolumn{8}{c}{\(\alpha\)} \\ \cline{3-11}  & & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1.0 \\ \hline \multirow{8}{*}{5} & 0.1 & 5.079 & 5.078 & 5.076 & 5.073 & 5.069 & 5.067 & 5.064 & 5.061  
AIB-GWMA-t chart at \(q=0.9\) is 3.047, which is close to 3.046 at \(\lambda=0.1\) for the existing AIB-EWMA-t chart, as shown by [PERSON] et al. (2018).