## Appendix A Confidence Intervals for Weight Parameters
### Parameter \(t\)  
General parameter \(\mathbf{\theta}\) can be determined as \(\mathbf{\theta}=\left(\mu\ \sigma^{2}\ t\right)^{T}\). Based on Equation (27), \(-\mathbf{\hat{J}}(\theta)\) can be expressed as  
\[-\mathbf{\hat{J}}(\theta)=\left( \begin{array}{ccc}\frac{\partial^{2}\mathbb{ \{\hat{\gamma}}}(\mathbf{\theta}|\mathbf{d}_{j})}{\partial\mu^{2}}&\frac{\partial^ {2}\mathbb{\{\hat{\gamma}}}(\mathbf{\theta}|\mathbf{d}_{j})}{\partial\mu\sigma^{2} }&\frac{\partial^{2}\mathbb{\{\hat{\gamma}}}(\mathbf{\theta}|\mathbf{d}_{j})}{ \partial\mu\sigma}\\ \frac{\partial^{2}\mathbb{\{\hat{\gamma}}}(\mathbf{\theta}|\mathbf{d}_{j})}{ \partial\
u^{2}\partial\mu}&\frac{\partial^{2}\mathbb{\{\hat{\gamma}}}(\mathbf{ \theta}|\mathbf{d}_{j})}{\partial\sigma^{2}\partial\mu}&\frac{\partial^{2} \mathbb{\{\hat{\gamma}}}(\mathbf{\theta}|\mathbf{d}_{j})}{\partial\sigma^{2} \partial\tau\sigma\sigma\sigma}\\ \frac{\partial^{2}\mathbb{\{\hat{\gamma}}}(\mathbf{\theta}|\mathbf{d}_{j})}{ \partial\delta\mu}&\frac{\partial^{2}\mathbb{\{\hat{\gamma}}}(\mathbf{\theta}| \mathbf{d}_{j})}{\partial\delta\sigma^{2}}&\frac{\partial^{2}\mathbb{\{ \hat{\gamma}}}(\mathbf{\theta}|\mathbf{d}_{j})}{\partial\delta\sigma^{2}}\end{array} \right) \tag{12}\]  
Based on the log-likelihood function in Equation (21), the first derivatives of the log-likelihood function \(\mathbb{\hat{\gamma}}(\mathbf{\theta}|\mathbf{d}_{j})\) for the mean, variance, and parameter \(t\) can be expressed as follows:  
\[ \begin{array}{c}\frac{\partial^{2}\mathbb{\{\hat{\gamma}}}(\mathbf{\theta}| \mathbf{d}_{j})}{\partial\mu}=\frac{\sum_{p=1}^{q}\left(d_{p}-\mu\right)}{ \sigma^{2}},\ \frac{\partial^{2}\mathbb{\{\hat{\gamma}}}(\mathbf{\theta}|\mathbf{d}_{j})}{ \partial\sigma^{2}}=-\frac{q}{2}\frac{1}{\sigma^{2}}+\frac{\sum_{p=1}^{q} \left(d_{p}-\mu\right)^{2}}{\frac{1}{\left(\sigma^{2}\right)^{2}}}\text{ and}\ \frac{\partial^{2}\mathbb{\{\hat{\gamma}}}(\mathbf{\theta}|\mathbf{d}_{j})}{ \partial t}=\frac{q}{t}+\sum_{p=1}^{q}log\left(\frac{y_{p}-U_{p}}{I_{p}-U_{p}} \right).\end{array}  \tag{13}\]  
The second derivatives of log-likelihood function \(\mathbb{\hat{\gamma}}(\mathbf{\theta}|\mathbf{d}_{j})\) for the mean, variance, and parameter \(t\) with mixed partials are given as follows:  
\[ \begin{array}{c}\frac{\partial^{2}\mathbb{\{\hat{\gamma}}}(\mathbf{\theta}| \mathbf{d}_{j})}{\partial\mu^{2}}=-\frac{q}{\sigma^{2}},\ \frac{\partial^{2}\mathbb{\{\hat{\gamma}}}(\mathbf{\theta}|\mathbf{d}_{j})}{ \partial\mu\sigma^{2}}=-\frac{\sum_{p=1}^{q}\left(d_{p}-\mu\right)}{\left(\sigma ^{2}\right)^{2}},\ \frac{\partial^{2}\mathbb{\{\hat{\gamma}}}(\mathbf{\theta}|\mathbf{d}_{j})}{ \partial\mu\sigma^{2}\partial\mu}=0,\ \frac{\partial^{2}\mathbb{\{\hat{\gamma}}}(\mathbf{\theta}|\mathbf{d}_{j})}{ \partial\sigma^{2}\partial\mu}=-\frac{\sum_{p=1}^{q}\left(d_{p}-\mu\right)}{\left( \sigma^{2}\right)^{2}},\\ \frac{\partial^{2}\mathbb{\{\hat{\gamma}}}(\mathbf{\theta}|\mathbf{d}_{j})}{\partial \left(\sigma^{2}\right)^{2}}=\frac{q}{2}\frac{1}{\left(\sigma^{2}\right)^{2}}- \frac{\sum_{p=1}^{q}\left(d_{p}-\mu\right)^{2}}{\left(\sigma^{2}\right)^{2}},\ \frac{\partial^{2}\mathbb{\{\hat{\gamma}}}(\mathbf{\theta}|\mathbf{d}_{j})}{ \partial\sigma^{2}\partial\tau}=0,\ \frac{\partial^{2}\mathbb{\{\hat{\gamma}}}(\mathbf{\theta}|\mathbf{d}_{j})}{ \partial\sigma^{2}\partial t}=0,\ \frac{\partial^{2}\mathbb{\{\hat{\gamma}}}(\mathbf{\theta}|\mathbf{d}_{j})}{ \partial\delta\sigma^{2}}=0\text{ and}\ \frac{\partial^{2}\mathbb{\{\hat{\gamma}}}(\mathbf{\theta}|\mathbf{d}_{j})}{ \partial\delta\sigma^{2}}=-\frac{q}{t^{2}}.\end{array}  \tag{14}\]  
Therefore,  
\[-\mathbf{\hat{J}}(\theta)=\left( \begin{array}{ccc}-\frac{q}{\sigma^{2}}&-\frac{ \sum_{p=1}^{q}\left(d_{p}-\mu\right)}{\left(\sigma^{2}\right)^{2}}&0\\ -\frac{\sum_{p=1}^{q}\left(d_{p}-\mu\right)}{\left(\sigma^{2}\right)^{2}}&\frac{q} {2}\frac{1}{\left(\sigma^{2}\right)^{2}}-\frac{\sum_{p=1}^{q}\left(d_{p}-\mu \right)^{2}}{\left(\sigma^{2}\right)^{3}}&0\\ 0&0&-\frac{q}{t^{2}}\end{array} \right)=\left(