## 2 Monitoring Process Precision and Accuracy  
The evaluation of process capability must take place in a stable statistical process. This study first constructs \(\delta-\gamma\) control charts to oversee process accuracy and precision. Next, this study derives a 100 (\(1-\alpha\))% confidence region of \((\delta,\gamma)\) from the in-control data. Let \(\left(Y_{i,1},\ldots,Y_{1,i},\ldots,Y_{1,n}\right)\) be the \(i\)th subsample, \(i=1\), \(2\), \(\ldots\), \(m\). According to the concept of [PERSON] [21], to estimate the unknown parameters \(-\delta\) and \(\gamma\), at least 20 to 25 subsamples should be taken. Then, the \(j\)th subsample and its observation values are shown below:  
\[\left(X_{j,1},\ldots,X_{j,i},\ldots,X_{j,n}\right)=\left(x_{j,1},\ldots,x_{j,i },\ldots,x_{j,n}\right) \tag{7}\]  
where \(j=1,2,\ldots,m\). Since \(Y_{j,i}=\left(X_{j,i}-T\right)/d\), then the \(j\)th subsample and its observation values processed by the variable transformation are written as follows:  
\[\left(Y_{j,1},\ldots,Y_{j,i},\ldots,Y_{j,n}\right)=\left(y_{j,1},\ldots,y_{j, i},\ldots,y_{j,n}\right) \tag{8}\]  
Accordingly, for the \(j\)th subsample, \(j=1,2,\ldots,m\), the subsample mean and the subsample variance are separately represented below:  
\[\hat{\delta}_{j}=\overline{Y}_{j}=\frac{1}{n}\sum_{i=1}^{n}Y_{j,i} \tag{9}\]  
\[\hat{\gamma}_{j}=S_{j}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(Y_{j,i}-\overline {Y}_{j}\right)} \tag{10}\]  
As normality is assumed, the \(j\)th subsample mean, \(\hat{\delta}_{j}\), is denoted as a normal distribution with mean \(\delta\) and standard deviation \(\gamma/\sqrt{n}\). The expected value of \(\hat{\delta}_{j}\) is \(\delta\) and the standard deviation of \(\hat{\delta}_{j}\) is \(\gamma/\sqrt{n}\). Obviously, \(\hat{\delta}_{j}\) is the unbiased estimator of accuracy index \(\delta\). Let \(K=n\hat{\gamma}_{j}^{2}/\gamma^{2}\), then \(K\) is regarded as a chi-square distribution with \(n-1\) degrees of freedom. The expected value of \(\hat{\gamma}_{j}\) is  
\[ \begin{array}{ll}E\left[\hat{\gamma}_{j}\right]&=E\left[\frac{\gamma}{\sqrt{n }}K^{1/2}\right]=\frac{\gamma}{\sqrt{n}}\int\limits_{0}^{\infty}k^{1/2}\frac{ 1}{\Gamma((n-1)/2)^{2(n-1)/2}}k^{(n-1)/2-1}e^{-k/2}d_{k}\\ &=\frac{\gamma}{\sqrt{n}}\frac{\Gamma(n/2)^{2n/2}}{\Gamma((n-1)/2)^{2(n-1)/2}} \int\limits_{0}^{\infty}\frac{1}{\Gamma(n/2)^{2n/2}}k^{n/2-1}e^{-k/2}d_{k}\\ &=b_{n}\gamma\end{array}  \tag{11}\]  
where  
\[b_{n}=\left(\frac{\sqrt{2}\Gamma(n/2)}{\sqrt{n}\Gamma((n-1)/2)}\right) \tag{12}\]  
Obviously, \(b_{n}^{-1}\hat{\gamma}_{j}\) is the unbiased estimator of precision index \(\gamma\). Furthermore, since \(E\left[\hat{\gamma}_{j}^{2}\right]=(n-1)/n\gamma^{2}\), then the standard deviation of \(\hat{\gamma}_{j}\) is  
\[\sigma_{\hat{\gamma}_{j}}=\sqrt{E\left[\hat{\gamma}_{j}^{2}\right]-E^{2}\left[ \hat{\gamma}_{j}\right]}=\sqrt{(n-1)/n-b_{n}^{2}}\gamma. \tag{13}\]  
Consequently, on the three-sigma control chart of accuracy index \(\delta\), the upper control limit, the center limit, and the lower control limit are defined as follows:  
\[UCL_{\delta}=\delta+\frac{3}{\sqrt{n}}\gamma \tag{14}\]  
\[CL_{\delta}=\delta \tag{15}\]  
\[LCL_{\delta}=\delta-\frac{3}{\sqrt{n}}\gamma. \tag{16}\]  
Similarly, on the three-sigma control chart of precision index \(\gamma\), the upper control limit, the center limit, and the lower control limit are represented as follows:  
\[UCL_{\gamma}=\left(b_{n}+3\sqrt{(n-1)/n-b_{n}^{2}}\right)\gamma \tag{17}\]  
\[CL_{\gamma}=b_{n}\gamma \tag{18}\]  
\[LCL_{\gamma}=\left(b_{n}-3\sqrt{(n-1)/n-b_{n}^{2}}\right)\gamma. \tag{19}\]  
Since the three control limits on the two control charts contain unknown parameters \(-\delta\) and \(\gamma\).