## 3 Frequentist Estimation
### Asymptotic Confidence Intervals  
To obtain the ACIs of the unknown parameters \(a_{j},j=1,2,\ldots,s\) and \(b\), we use the Fisher information matrix introduced in [23], and the matrix is derived by calculating the corresponding second-order partial derivatives. Then, we present the Fisher information matrix as:  
\[\frac{\partial^{2}\ln L(\mathbf{a},b)}{\partial a_{k}\partial a_{j}}= \begin{cases} \frac{-m_{j}}{a_{j}^{2}}+\frac{n_{2}}{a^{*2}}-B&\text{if }k=j\\ \frac{n_{2}}{a^{*2}}-B&\text{if }k\
eq j\end{cases} k,j=1,2,\ldots,s, \tag{27}\]  
where  
\[B=\sum_{i=n_{1}+1}^{n_{1}+n_{2}}\frac{\psi_{r_{i}}^{-a^{*}}\psi_{l_{i}}^{-a^{ *}}[\ln\psi_{r_{i}}-\ln\psi_{l_{i}}]^{2}}{(\psi_{l_{i}}^{-a^{*}}-\psi_{r_{i}}^ {-a^{*}})^{2}},\]  
\(\psi_{l_{i}}=1+l_{i}^{b}\), and \(\psi_{r_{i}}=1+r_{i}^{b}\). In addition,  
\[\frac{\partial^{2}\ln L(\mathbf{a},b)}{\partial a_{j}\partial b}=-\sum_{i=1}^ {n_{1}}\frac{t_{i}^{b}\ln t_{i}}{1+t_{i}^{b}}+C+D,j=1,2,\ldots,s, \tag{28}\]  
where  
\[C=\sum_{i=n_{1}+1}^{n_{1}+n_{2}}\frac{r_{i}^{b}\psi_{r_{i}}^{-(a^{*}+1)}\ln r _{i}-l_{i}^{b}\psi_{l_{i}}^{-(a^{*}+1)}\ln l_{i}}{\psi_{l_{i}}^{-a^{*}}-\psi_{ r_{i}}^{-a^{*}}},\]  
\[D=a^{*}\sum_{i=n_{1}+1}^{n_{1}+n_{2}}(\psi_{r_{i}}\psi_{l_{i}})^{-(a^{*}+1)}( \ln\psi_{r_{i}}-\ln\psi_{l_{i}})\frac{\psi_{r_{i}}l_{i}^{b}\ln l_{i}-\psi_{l_ {i}}r_{i}^{b}\ln r_{i}}{(\psi_{l_{i}}^{-a^{*}}-\psi_{r_{i}}^{-a^{*}})^{2}}.\]  
For the second-order derivative of \(b\),  
\[\frac{\partial^{2}\ln L(\mathbf{a},b)}{\partial b^{2}}=-\frac{n_{1}}{b^{2}}-(a ^{*}+1)\sum_{i=1}^{n_{1}}\frac{t_{i}^{b}(\ln t_{i})^{2}}{(1+t_{i}^{b})^{2}}+E -F, \tag{29}\]  
where\[E=a^{*}\sum_{i=n_{1}+1}^{n_{1}+n_{2}}\frac{\psi_{l_{i}}^{-(a^{*}+2)}l_{i}^{b}(\ln l _{i})^{2}(a^{*}l_{i}^{b}-1)-\psi_{r_{i}}^{-(a^{*}+2)}r_{i}^{b}(\ln r_{i})^{2}(a^{* }r_{i}^{b}-1)}{\psi_{l_{i}}^{-a^{*}}-\psi_{r_{i}}^{-a^{*}}},\]  
\[F={a^{*}}^{2}\sum_{i=n_{1}+1}^{n_{1}+n_{2}}\frac{[\psi_{l_{i}}^{-(a^{*}+1)}l_{i} ^{b}\ln l_{i}-\psi_{r_{i}}^{-(a^{*}+1)}l_{i}^{b}\ln r_{i}]^{2}}{(\psi_{l_{i}}^{ -a^{*}}-\psi_{r_{i}}^{-a^{*}})^{2}}.\]  
For simplicity, the Fisher matrix is given for \(s=2\), and the form for other values of \(s\) can be obtained similarly. In addition, we denote \(\ln L(\mathbf{a},b)\) by \(\ln L\) in the following matrix. The Fisher information matrix is given by:  
\[I(a_{1},a_{2},b)=-E\left[ \begin{array}{ccc}\frac{\partial^{2}\ln L}{ \partial a_{1}^{2}}&\frac{\partial^{2}\ln L}{\partial a_{1}\partial a_{2}}& \frac{\partial^{2}\ln L}{\partial a_{1}\partial b}\\ \frac{\partial^{2}\ln L}{\partial a_{2}\partial a_{1}}&\frac{\partial^{2}\ln L }{\partial a_{2}^{2}\partial a_{2}\partial b^{2}}\\ \frac{\partial^{2}\ln L}{\partial b\partial a_{1}}&\frac{\partial^{2}\ln L}{ \partial b\partial a_{2}}&\frac{\partial^{2}\ln L}{\partial b^{2}}\\ \end{array} \right].\]  
Replace the unknown parameters by their MLEs, then the observed Fisher matrix \(I_{o}(\hat{a_{1}},\hat{a_{2}},\hat{b})\) can be given as:  
\[I_{o}(\hat{a_{1}},\hat{a_{2}},\hat{b})=-\left[ \begin{array}{ccc}\frac{ \partial^{2}\ln L}{\partial a_{1}^{2}}&\frac{\partial^{2}\ln L}{\partial a_{ 1}\partial a_{2}}&\frac{\partial^{2}\ln L}{\partial a_{1}\partial b}\\ \frac{\partial^{2}\ln L}{\partial a_{2}\partial a_{1}}&\frac{\partial^{2}\ln L }{\partial a_{2}^{2}}&\frac{\partial^{2}\ln L}{\partial a_{2}\partial b}\\ \frac{\partial^{2}\ln L}{\partial b\partial a_{1}}&\frac{\partial^{2}\ln L}{ \partial b\partial a_{2}}&\frac{\partial^{2}\ln L}{\partial b^{2}}\\ \end{array} \right]_{(a_{1},a_{2},b)=(\hat{a_{1}},\hat{a_{2}},\hat{b})}.\]  
We define \(V(\hat{a_{1}},\hat{a_{2}},\hat{b})\) as the inverse matrix of \(I_{o}(\hat{a_{1}},\hat{a_{2}},\hat{b})\), which is expressed as:  
\[V(\hat{a_{1}},\hat{a_{2}},\hat{b})=I_{o}^{-1}(\hat{a_{1}},\hat{a_{2}},\hat{b}) =\left[ \begin{array}{ccc}var(\hat{a_{1}})&cor(\hat{a_{1}},\hat{a_{2}})&cov( \hat{a_{1}},\hat{b})\\ cov(\hat{a_{1}},\hat{a_{2}})&var(\hat{a_{2}})&cov(\hat{a_{2}},\hat{b})\\ cov(\hat{a_{1}},\hat{b})&cov(\hat{a_{2}},\hat{b})&var(\hat{b})\\ \end{array}