## Appendix A Confidence Intervals for Weight Parameters
### Parameter \(r\)  
To estimate parameter\(r\), general parameter \(\mathbf{\theta}\) can be determined as \(\mathbf{\theta}=\big{(}\mu\;\sigma^{2}\;r\big{)}^{T}\). From Equation (27),  
\[-\hat{\mathbf{J}}(\mathbf{\theta})=\left( \begin{array}{cc}\frac{\partial^{2}\mathbb{ I}(\mathbf{\theta}|\mathbf{d}_{j})}{\partial\mu^{2}}&\frac{\partial^{2}\mathbb{ I}(\mathbf{\theta}|\mathbf{d}_{j})}{\partial\mu\partial\sigma^{2}}&\frac{\partial^{2} \mathbb{I}(\mathbf{\theta}|\mathbf{d}_{j})}{\partial\mu\partial\sigma}\\ \frac{\partial^{2}\mathbb{I}(\mathbf{\theta}|\mathbf{d}_{j})}{\partial\mu^{2} \partial\mu}&\frac{\partial^{2}\mathbb{I}(\mathbf{\theta}|\mathbf{d}_{j})}{ \partial(\sigma^{2})^{2}}&\frac{\partial^{2}\mathbb{I}(\mathbf{\theta}|\mathbf{d} _{j})}{\partial\sigma^{2}\partial\tau}\\ \frac{\partial^{2}\mathbb{I}(\mathbf{\theta}|\mathbf{d}_{j})}{\partial\sigma\mu \partial\mu}&\frac{\partial^{2}\mathbb{I}(\mathbf{\theta}|\mathbf{d}_{j})}{ \partial\sigma\sigma^{2}}&\frac{\partial^{2}\mathbb{I}(\mathbf{\theta}|\mathbf{d} _{j})}{\partial\tau^{2}}\end{array} \right) \tag{12}\]  
Based on the log-likelihood function in Equation (23), the first derivatives of the log-likelihood function \(\mathbb{I}(\mathbf{\theta}|\mathbf{d}_{j})\) for the mean, variance, and parameter \(r\) can be expressed as follows:  
\[\frac{\partial^{2}\mathbb{I}(\mathbf{\theta}|\mathbf{d}_{j})}{\partial\mu}=\frac{ \sum_{p=1}^{q}\big{(}d_{p}-\mu\big{)}}{\sigma^{2}},\;\frac{\partial^{2} \mathbb{I}(\mathbf{\theta}|\mathbf{d}_{j})}{\partial\sigma^{2}}=-\frac{q}{2}\frac {1}{\sigma^{2}}+\frac{\sum_{p=1}^{q}\big{(}d_{p}-\mu\big{)}^{2}}{2}\frac{1}{( \sigma^{2})^{2}}\;\text{and}\;\frac{\partial^{2}\mathbb{I}(\mathbf{\theta}|\mathbf{ d}_{j})}{\partial\tau}=\frac{q}{\tau}+\sum_{p=1}^{q}log\Big{(}\frac{y_{p}-U_{p}}{ L_{p}-U_{p}}\Big{)}. \tag{13}\]  
The second derivatives of log-likelihood function \(\mathbb{I}(\mathbf{\theta}|\mathbf{d}_{j})\) for the mean, variance, and parameter \(r\) with mixed partials are given as follows:  
\[ \begin{array}{c}\frac{\partial^{2}\mathbb{I}(\mathbf{\theta}|\mathbf{d}_{j})}{ \partial\mu^{2}}=-\frac{q}{\sigma^{2}},\;\frac{\partial^{2}\mathbb{I}(\mathbf{ \theta}|\mathbf{d}_{j})}{\partial\mu\partial\sigma^{2}}=-\frac{\sum_{p=1}^{q} \big{(}d_{p}-\mu\big{)}}{(\sigma^{2})^{2}},\;\frac{\partial^{2}\mathbb{I}(\bm {\theta}|\mathbf{d}_{j})}{\partial\mu\partial\sigma^{2}}=0,\;\frac{\partial^{2} \mathbb{I}(\mathbf{\theta}|\mathbf{d}_{j})}{\partial\sigma^{2}\partial\mu}=-\frac{ \sum_{p=1}^{q}\big{(}d_{p}-\mu\big{)}}{(\sigma^{2})^{2}},\\ \frac{\partial^{2}\mathbb{I}(\mathbf{\theta}|\mathbf{d}_{j})}{\partial(\sigma^{2})^ {2}}=\frac{q}{2}\frac{1}{(\sigma^{2})^{2}}-\frac{\sum_{p=1}^{q}\big{(}d_{p}- \mu\big{)}^{2}}{(\sigma^{2})^{3}},\;\frac{\partial^{2}\mathbb{I}(\mathbf{\theta}| \mathbf{d}_{j})}{\partial\sigma^{2}\sigma}=0,\;\frac{\partial^{2}\mathbb{I}( \mathbf{\theta}|\mathbf{d}_{j})}{\partial\sigma\mu\mu}=0,\;\frac{\partial^{2} \mathbb{I}(\mathbf{\theta}|\mathbf{d}_{j})}{\partial\sigma\partial\sigma^{2}}=0 \;\text{and}\;\frac{\partial^{2}\mathbb{I}(\mathbf{\theta}|\mathbf{d}_{j})}{ \partial\tau^{2}}=-\frac{q}{r^{2}}.\end{array}  \tag{14}\]  
Therefore,  
\[-\hat{\mathbf{J}}(\mathbf{\theta})=\left( \begin{array}{cc}-\frac{q}{\sigma^{2}}&-\frac{ \sum_{p=1}^{q}\big{(}d_{p}-\mu\big{)}}{(\sigma^{2})^{2}}&0\\ -\frac{\sum_{p=1}^{q}\big{(}d_{p}-\mu\big{)}}{(\sigma^{2})^{2}}&\frac{q}{2}\frac {1}{(\sigma^{2})^{2}}-\frac{\sum_{p=1}^{q}\big{(}d_{p}-\mu\big{)}^{2}}{( \sigma^{2})^{3}}&0\\ 0&0&-\frac{q}{r^{2}}\end{array} \right)=\left( \begin{array}{cc}\hat{\mathbf{J}}( \mathbf{\theta})_{11}&\hat{\mathbf{J}}(\mathbf{\theta})_{12}&\hat{\mathbf{J}}(\mathbf{\theta})_{13} \\ \hat{\mathbf{J}}(\mathbf{\theta})_{21}&\hat{\mathbf{J}}(\mathbf{\theta})_{22}&\hat{\mathbf{J}}(\mathbf{ \theta})_{23}\\ \hat{\mathbf{J}}(\mathbf{\theta})_{31}&\hat{\mathbf{J}}(\mathbf{\theta})_{32}&\hat{\mathbf{J}}(\mathbf{ \theta})_{33}\end{array}