## 6 Research Limitations and Future Research
Therefore,  
\[p\!\left(-t_{\alpha/4\mu-1}\leq\frac{\sqrt{n}\big{(}\delta^{\prime}_{h}- \delta^{\prime}_{h}\big{)}}{\hat{\gamma}_{h}}\leq t_{\alpha/4\mu-1}\right)=p \big{(}\delta^{\prime}_{h}-e_{h}\leq\delta^{\prime}_{h}\leq\delta^{\prime}_{h} +e_{h}\big{)}=1-\frac{\alpha}{2}, \tag{15}\]  
where \(e_{h}=t_{\alpha/4\mu-1}\hat{\gamma}_{h}/\sqrt{n}\) and \(t_{\alpha/4\mu-1}\) is the upper \(\alpha/4\) quintile of \(t_{n-1}\). Similarly,  
\[p\!\left(\chi^{2}_{\alpha/4\mu-1}\leq\frac{n\hat{\gamma}^{2}_{h}}{\gamma^{2}_ {h}}\leq\chi^{2}_{1-\alpha/4\mu-1}\right)=p\!\left(\sqrt{\frac{n}{\chi^{2}_{1- \alpha/4\mu-1}}}\hat{\gamma}^{\prime}_{h}\leq\gamma^{\prime}_{h}\leq\sqrt{ \frac{n}{\chi^{2}_{\alpha/4\mu-1}}}\hat{\gamma}^{\prime}_{h}\right)=1-\frac{ \alpha}{2}, \tag{16}\]  
where \(\chi^{2}_{\alpha/4\mu-1}\) is the lower \(\alpha/4\) quintile of \(\chi^{2}_{n-1}\), and \(\chi^{2}_{1-\alpha/4\mu-1}\) is the lower \(1-\alpha/4\) quintile of \(\chi^{2}_{n-1}\). Obviously, when \(\delta^{\prime}_{h}-e_{h}\geq 0\), it can be inferred that \(\delta^{\prime}_{h}\geq 0\) and \(\mu_{h}\geq T_{h}\). Likewise, when \(\delta^{\prime}_{h}+e_{h}<0\), it can be inferred that \(\delta^{\prime}_{h}<0\) and \(\mu_{h}<T_{h}\). Thus,  
\[\delta_{h}=\left\{ \begin{array}{l}\hat{\delta}^{\prime}_{h}d_{h1},\hat{ \delta}^{\prime}_{h}-e_{h}\geq 0\\ \hat{\delta}^{\prime}_{h}d_{h2},\hat{\delta}^{\prime}_{h}+e_{h}<0\end{array} .\right. \tag{17}\]  
In order to derive the \(100(1-\alpha)\%\) confidence region of \(\big{(}\delta^{\prime}_{h},\gamma^{\prime}_{h}\big{)}\), this paper defines events \(E_{\delta h}\) and \(E_{\gamma h}\) as follows:  
\[E_{\delta h}=\big{\{}\delta^{\prime}_{Lh}\leq\delta^{\prime}_{h}\leq\delta^{ \prime}_{Uh}\big{\}}=\left\{\delta^{\prime}_{h}-\frac{t_{\alpha/4\mu-1}}{ \sqrt{n}}\hat{\gamma}^{\prime}_{h}\leq\delta^{\prime}_{h}\leq\delta^{\prime}_{ h}+\frac{t_{\alpha/4\mu-1}}{\sqrt{n}}\hat{\gamma}^{\prime}_{h}\right\} \tag{18}\]  
\[E_{\gamma h}=\big{\{}\gamma^{\prime}_{Lh}\leq\gamma^{\prime}_{h}\leq\gamma^{ \prime}_{Uh}\big{\}}=\left\{\sqrt{\frac{n}{\chi^{2}_{1-\alpha/4\mu-1}}}\hat{ \gamma}^{\prime}_{h}\leq\gamma^{\prime}_{h}\leq\sqrt{\frac{n}{\chi^{2}_{ \alpha/4\mu-1}}}\hat{\gamma}^{\prime}_{h}\right\}. \tag{19}\]  
Obviously, \(p(E_{\delta h})=p\big{(}E_{\gamma h}\big{)}=1-\alpha/2\) and \(p\big{(}E_{\delta h}^{\ast}\big{)}=p\Big{(}E_{\gamma h}^{C}\big{)}=\alpha/2\). Based on Boole's inequality and DeMorgan's rules, we have \(p\big{(}E_{\delta h}\cap\Sigma_{\gamma h}\big{)}\geq 1-p\big{(}E_{\delta h}^{C} \big{)}-p\Big{(}E_{\gamma h}^{C}\big{)}=\alpha/2\). Thus,  
\[p\!\left\{\delta^{\prime}_{h}-\frac{t_{\alpha/4\mu-1}}{\sqrt{n}}\hat{\gamma}^{ \prime}_{h^{\prime}}\leq\delta^{\prime}_{h}\leq\hat{\delta}^{\prime}_{h}+\frac {t_{\alpha/4\mu-1}}{\sqrt{n}}\hat{\gamma}^{\prime}_{h^{\prime}}\sqrt{\frac{n}{ \chi^{2}_{1-\alpha/4\mu-1}}}\hat{\gamma}^{\prime}_{h}\leq\gamma^{\prime}_{h} \leq\sqrt{\frac{n}{\chi^{2}_{\alpha/4\mu-1}}}\hat{\gamma}^{\prime}_{h}\right\}=1-\alpha. \tag{20}\]Based on Equation (A8), \(\delta_{h}=\delta_{h}^{\prime}d_{h1}\) for \(\delta_{Lh}^{\prime}\geq 0\), then  
\[ \begin{array}{l}p\Big{\{}\delta_{h}^{\prime}\times d_{h1}-\frac{t_{a/4m-1}}{ \sqrt{n}}\gamma_{h}^{\prime}\times d_{h1},\leq\delta_{h}\leq\delta_{h}^{\prime }\times d_{h1}+\frac{t_{a/4m-1}}{\sqrt{n}}\hat{\gamma}_{h}\times d_{h1}\\ \sqrt{\overline{x_{1-a/4m-1}^{2}}}\gamma_{h}^{\prime}\times d_{h1}\leq\gamma_ {h}\leq\sqrt{\overline{x_{a/4m-1}^{2}}}\gamma_{h}^{\prime}\times d_{h1}\Big{\}} =1-\alpha,\end{array}  \tag{12}\]  
where \(\hat{\gamma}_{h}=\gamma_{h}^{\prime}d_{h1}\), \(\delta_{h}=\delta_{h}^{\prime}d_{h1}\). Therefore, \(100(1-\alpha)\%\) confidence region of \((\delta_{h},\gamma_{h})\) for \(\delta_{Lh}^{\prime}\geq 0\) is expressed as:  
\[ \begin{array}{l}CR=\big{\{}\left(\delta_{h},\gamma_{h}\right)|\delta_{Lh}^{ \prime}\times d_{h1}\leq\delta_{h}\leq\delta_{UH}^{\prime}\times d_{h1}, \gamma_{Lh}^{\prime}\times d_{h1}\leq\gamma_{h}\leq\gamma_{UH}^{\prime}\times d _{h1}\big{\}}.\end{array}