## 3 Frequentist Estimation
### EM Algorithm  
Given that explicit expressions for \(\hat{a}_{j}\) and \(\hat{b}\) are unavailable, the EM algorithm is considered and employed to find the MLEs. Incomplete data scenarios are typically addressed using the EM algorithm, which is first formulated by [PERSON] et al. [22]. Define \(t_{i}^{*}\) as the exact survival time of the unit that falls within the interval \([l_{i},\,r_{i}]\), \(i=n_{1}+1,\ldots,\,n_{1}+n_{2}\). Based on the complete data \((t_{1},t_{2},\ldots,t_{n_{1}},t_{n_{1}+1}^{*},\ldots,t_{n_{1}+n_{2}}^{*})\), we can derive the log-likelihood function, which is expressed as:  
\[ \begin{split}\ln L_{\text{C}}(\mathbf{a},b)&=\sum \limits_{j=1}^{s}m_{j}\ln a_{j}+n\ln b+(b-1)(\sum\limits_{i=1}^{n_{1}}\ln t_{ i}+\sum\limits_{i=n_{1}+1}^{n_{1}+n_{2}}\ln t_{i}^{*})\\ &\qquad-(a^{*}+1)(\sum\limits_{i=1}^{n_{1}}\ln\left(1+t_{i}^{b} \right)+\sum\limits_{i=n_{1}+1}^{n_{1}+n_{2}}\ln\left(1+t_{i}^{*b}\right)), \end{split}  \tag{20}\]  
then by setting the derivatives of \(\ln L_{\text{C}}(\mathbf{a},b)\) with respect to \(a_{j}\) and \(b\) to \(0\), we can obtain:  
\[\frac{\partial\ln L_{\text{C}}(\mathbf{a},b)}{\partial a_{j}}=\frac{m_{j}}{a_ {j}}-(\sum\limits_{i=1}^{n_{1}}\ln\left(1+t_{i}^{b}\right)+\sum\limits_{i=n_{ 1}+1}^{n_{1}+n_{2}}\ln\left(1+t_{i}^{*b}\right))=0,j=1,2,\ldots,s, \tag{21}\]  
\[ \begin{split}\frac{\partial\ln L_{\text{C}}(\mathbf{a},b)}{ \partial b}=\frac{n}{b}+(\sum\limits_{i=1}^{n_{1}}\ln t_{i}+\sum\limits_{i=n_{ 1}+1}^{n_{1}+n_{2}}\ln t_{i}^{*})\\ -(a^{*}+1)(\sum\limits_{i=1}^{n_{1}}\frac{t_{i}^{b}\ln t_{i}}{1+t _{i}^{b}}+\sum\limits_{i=n_{1}+1}^{n_{1}+n_{2}}\frac{(t_{i}^{b})^{b}\ln t_{i}^ {*}}{1+t_{i}^{*b}})=0.\end{split}  \tag{22}\]Through Equation (21), we can obtain that  
\[\frac{m_{j}}{d_{j}}=\frac{n}{\hat{a^{*}}},j=1,2,\ldots,s, \tag{23}\]  
and we can derive the system of equations:  
\[ \begin{cases}\frac{n}{a^{*}}-(\sum_{i=1}^{n_{1}}\ln{(1+t_{i}^{b})}+\sum_{i=n_{1} +1}^{n_{1}+n_{2}}\ln{(1+t_{i}^{*b})})=0\\ \frac{n}{b}+(\sum_{i=1}^{n_{1}}\ln{t_{i}}+\sum_{i=n_{1}+1}^{n_{1}+n_{2}}\ln{t_ {i}^{*}})-(a^{*}+1)(\sum_{i=1}^{n_{1}}\frac{t_{i}^{b}\ln{t_{i}}}{1+t_{i}^{b}}+ \sum_{i=n_{1}+1}^{n_{1}+n_{2}}\frac{t_{i}^{b}\ln{t_{i}^{*}}}{1+t_{i}^{b}})=0 \end{cases} . \tag{24}\]  
For conciseness, we define the conditional expectations as follows:  
\[E_{\mathbf{a},b}(\ln{T_{i}}|T_{i}\in[l_{i},r_{i}],D_{i}=d_{i})=\int_{l_{i}}^{ r_{i}}\frac{\ln{t}\times f(t,d_{i};\mathbf{a},b)}{F(r_{i},d_{i};\mathbf{a},b)-F(l_{i},d_{i};\mathbf{a},b)}dt,\]  
\[E_{\mathbf{a},b}(\ln{(1+T_{i}^{b})}|T_{i}\in[l_{i},r_{i}],D_{i}=d_{i})=\int_{l _{i}}^{r_{i}}\frac{\ln{(1+t^{b})}\times f(t,d_{i};\mathbf{a},b)}{F(r_{i},d_{i} ;\mathbf{a},b)-F(l_{i},d_{i};\mathbf{a},b)}dt,\]  
\[E_{\mathbf{a},b}(\frac{T_{i}^{b}\ln{T_{i}}}{1+T_{i}^{b}}|T_{i}\in[l_{i},r_{i}],D_{i}=d_{i})=\int_{l_{i}}^{r_{i}}\frac{t^{b}\ln{t}}{1+t^{b}}\times\frac{f(t,d _{i};\mathbf{a},b)}{F(r_{i},d_{i};\mathbf{a},b)-F(l_{i},d_{i};\mathbf{a},b)}dt,\]  
where the expressions of \(f(t,d_{i};\mathbf{a},b)\) and \(F(t,d_{i};\mathbf{a},b)\) can be seen in (7) and (8). In each iteration, to perform the E-step, we need to calculate the expectations using the parameter values obtained from the previous iteration by numerical integration methods. For notational convenience, we introduce the symbols as follows:  
* \(E_{1,i}^{(k)}=E_{\mathbf{a}^{(k-1)},b^{(k-1)}}(\ln{T_{i}}|T_{i}\in[l_{i},r_{i }],D_{i}=d_{i})\),
* \(E_{2,i}^{(k)}=E_{\mathbf{a}^{(k-1)},b^{(k-1)}}(\ln{(1+T_{i}^{b})}|T_{i}\in[l_ {i},r_{i}],D_{i}=d_{i})\),
* \(E_{3,i}^{(k)}=E_{\mathbf{a}^{(k-1)},b^{(k-1)}}(\frac{T_{i}^{b}\ln{T_{i}}}{1+T_ {i}^{b}}|T_{i}\in[l_{i},r_{i}],D_{i}=d_{i})\),  
where \(i=n_{1}+1,n_{1}+2,\ldots,n_{1}+n_{2}\). And for those uncensored data, we define:  
\[F_{1,i}^{(k)}=\ln{t_{i}},F_{2,i}^{(k)}=\ln{(1+t_{i}^{b^{(k-1)}})},F_{3,i}^{( k)}=\frac{t_{i}^{b^{(k-1)}}\ln{t_{i}}}{1+t_{i}^{b^{(k-1)}}},\]  
where \(i=1,2,\ldots,n_{1}\).