## 4 Proposed Transfer Functions Based on DFs
### Confidence Intervals for Weight Parameters
#### 4.3.1 Confidence Intervals for Parameter s  
The general parameter \(\boldsymbol{\theta}\) can be determined as \(\boldsymbol{\theta}=\big{(}\mu\sigma^{2}\;s\big{)}^{T}\). From Equation (27), \(-\hat{\boldsymbol{J}}(\theta)\) can be given as  
\[-\hat{\boldsymbol{J}}(\theta)=\left(\begin{array}{ccc}\frac{\partial^{2} \updownarrow(\theta|\mathbf{d}_{j})}{\partial\mu^{2}}&\frac{\partial^{2} \updownarrow(\theta|\mathbf{d}_{j})}{\partial\mu^{2}}&\frac{\partial^{2} \updownarrow(\theta|\mathbf{d}_{j})}{\partial\mu^{2}}\\ \frac{\partial^{2}\updownarrow(\theta|\mathbf{d}_{j})}{\partial\
u^{2}\mu}& \frac{\partial^{2}\updownarrow(\theta|\mathbf{d}_{j})}{\partial(\sigma^{2})^ {2}}&\frac{\partial^{2}\updownarrow(\theta|\mathbf{d}_{j})}{\partial\
u^{2} \alpha\delta_{3}}\\ \frac{\partial^{2}\updownarrow(\theta|\mathbf{d}_{j})}{\partial\delta\mu}& \frac{\partial^{2}\updownarrow(\theta|\mathbf{d}_{j})}{\partial\delta\
u^{2} }&\frac{\partial^{2}\updownarrow(\theta|\mathbf{d}_{j})}{\partial\delta^{2}} \end{array}\right). \tag{31}\]  
From the log-likelihood function in Equation (17), \(-\hat{\boldsymbol{J}}(\theta)\) can be calculated as follows (see Appendix A.1):  
\[-\hat{\boldsymbol{J}}(\theta)=\left(\begin{array}{ccc}-\frac{q}{\sigma^{2} }&-\frac{\updownarrow_{p=1}^{q}\left(d_{p}-\mu\right)}{\left(\sigma^{2}\right) ^{2}}&0\\ -\frac{\updownarrow_{p=1}^{q}\left(d_{p}-\mu\right)}{\left(\sigma^{2}\right) ^{2}}&\frac{q}{2}\frac{1}{\left(\sigma^{2}\right)^{2}}-\frac{\updownarrow_{ p=1}^{q}\left(d_{p}-\mu\right)^{2}}{\left(\sigma^{2}\right)^{2}}&0\\ 0&0&-\frac{q}{\sigma^{2}}\end{array}\right)=\left(\begin{array}{ccc}\hat{ \boldsymbol{J}}(\theta)_{11}&\hat{\boldsymbol{J}}(\theta)_{12}&\hat{ \boldsymbol{J}}(\theta)_{13}\\ \hat{\boldsymbol{J}}(\theta)_{21}&\hat{\boldsymbol{J}}(\theta)_{22}&\hat{ \boldsymbol{J}}(\theta)_{23}\\ \hat{\boldsymbol{J}}(\theta)_{31}&\hat{\boldsymbol{J}}(\theta)_{32}&\hat{ \boldsymbol{J}}(\theta)_{33}\end{array}\right) \tag{32}\]  
The square root of the diagonal elements of \(\hat{\boldsymbol{J}}\big{(}\hat{\boldsymbol{\theta}}\big{)}^{-1}\) represents an estimate of the standard error for each estimator. An estimator for the standard error of \(\hat{s}\) is of particular interest and can be found using the inverses of the partitioned matrices. First, the partition \(-\hat{\boldsymbol{J}}(\theta)\) is as follows:  
\[-\hat{\boldsymbol{J}}(\theta)=\left(\begin{array}{cc}\hat{\boldsymbol{J}}( \theta)_{11}&\hat{\boldsymbol{J}}(\theta)_{12}\\ \hat{\boldsymbol{J}}(\theta)_{21}&\hat{\boldsymbol{J}}(\theta)_{22}\end{array}\right). \tag{33}\]  
Next, one estimator for \(\text{Var}(\hat{s})\) can be obtained as follows:  
\[\text{Var}(\hat{s})=\big{(}\hat{\boldsymbol{J}}(\theta)_{11}\hat{\boldsymbol{J} }(\theta)_{22}-\hat{\boldsymbol{J}}(\theta)_{21}\hat{\boldsymbol{J}}(\theta)_{1 2}\big{)}^{-1}\] \[=-\bigg{[}\frac{q}{\sigma^{2}}\bigg{(}\frac{q}{2}\frac{1}{(\sigma^ {2})^{2}}-\frac{\updownarrow_{p=1}^{q}\left(d_{p}-\mu\right)^{2}}{(\sigma^{2}) ^{3}}\bigg{)}+\bigg{(}\frac{\updownarrow_{p=1}^{q}\left(d_{p}-\mu\right)}{( \sigma^{2})^{2}}\bigg{)}^{2}\bigg{]}^{-1}. \tag{34}\]The associated standard error of \(\hat{s}\) can be calculated by  
\[\hat{\text{SE}}(\hat{s})=\frac{1}{\sqrt{-\left[\frac{\hat{q}}{\sigma^{2}}\left( \frac{\hat{q}}{2}\frac{1}{\left(\sigma^{2}\right)^{2}}-\frac{\sum_{p=1}^{q} \left(d_{p}-\mu\right)^{2}}{\left(\sigma^{2}\right)^{3}}\right)+\left(\frac{ \sum_{p=1}^{q}\left(d_{p}-\mu\right)}{\left(\sigma^{2}\right)^{2}}\right)^{2} \right]}}. \tag{35}\]  
Therefore, an approximation \((1-\alpha)100\%\) confidence interval of \(s\) can be obtained by using  
\[\hat{s}\pm z^{*}\hat{\text{SE}}(\hat{s}). \tag{36}\]