## 3 Frequentist Estimation
### Maximum Likelihood Estimators  
For simplicity, according to the approach in [8], after reordering the data, the sample is partitioned such that the first \(n_{1}\) data items are not censored, and the subsequent \(n_{2}\) data items are middle-censored. Thus, our data can be written as:\[\{(T_{1},D_{1}),\ldots,(T_{n_{1}},D_{n_{1}}),([L_{n_{1}+1},R_{n_{1}+1}],D_{n_{1}+1} ),\ldots,([L_{n_{1}+1},R_{n_{1}+n_{2}}],D_{n_{1}+n_{2}})\}.\]  
Let \(data\) represent the observed sample data in the form mentioned above. Based on the sample data, the likelihood function can be derived through (7) and (8) as follows:  
\[ \begin{split} L(\mathbf{a},b;data)\propto\prod_{j=1}^{s}\prod _{i=1}^{n_{1}}(f(t_{i},j;\mathbf{a},b))^{I(D_{i}=j)}\prod_{j=1}^{s}\prod_{i=n_ {1}+1}^{n_{1}+n_{2}}(F(r_{i},j;\mathbf{a},b)-F(l_{i},j;\mathbf{a},b))^{I(D_{i}= j)}\\ =b^{n_{1}}(\frac{1}{a^{*}})^{n_{2}}\prod_{j=1}^{s}a_{j}^{n_{j}} \prod_{i=1}^{n_{1}}[[t_{i}^{b-1}(1+t_{i}^{b})^{-(a^{*}+1)}]\prod_{i=n_{1}+1}^ {n_{1}+n_{2}}[(1+t_{i}^{b})^{-a^{*}}-(1+t_{i}^{b})^{-a^{*}}]\end{split}  \tag{9}\]  
where symbol \(\propto\) denotes 'proportional to',  
\[I(D_{i}=j){=}\left\{ \begin{array}{ll}1&\text{if }D_{i}=j,\\ 0&\text{Otherwise,}\end{array} \right.\]  
and \(m_{j}=\sum_{i=1}^{n}I(D_{i}=j)\), which equals the number of the items that have failed attributed to the \(j\)-th latent risk. For simplicity, we use \(L(\mathbf{a},b)\) to replace \(L(\mathbf{a},b;data)\). Through (9), the log-likelihood function takes the form:  
\[ \begin{split}\ln L(\mathbf{a},b)&=n_{1}\ln b-n_{2}\ln a ^{*}+\sum_{j=1}^{s}m_{j}\ln a_{j}+(b-1)\sum_{i=1}^{n_{1}}\ln t_{i}\\ -(a^{*}+1)\sum_{i=1}^{n_{1}}\ln{(1+t_{i}^{b})}+\sum_{i=n_{1}+1}^ {n_{1}+n_{2}}\ln{[(1+t_{i}^{b})^{-a^{*}}-(1+t_{i}^{b})^{-a^{*}}]}.\end{split}  \tag{10}\]  
To solve for the MLEs, we set the first-order partial derivatives of function \(\ln L(\mathbf{a},\mathbf{b})\) with respect to all unknown parameters \(a_{1},a_{2},\ldots,a_{s}\) and \(b\) to 0, respectively. Consequently, the corresponding likelihood equations are given by:  
\[\frac{\partial\ln L(\mathbf{a},b)}{\partial a_{j}}=\frac{m_{j}}{a_{j}}-\frac{ n_{2}}{a^{*}}-\sum_{i=1}^{n_{1}}\ln{(1+t_{i}^{b})}+K_{1}=0,j=1,2,3,\ldots,s, \tag{11}\]  
\[\frac{\partial\ln L(\mathbf{a},b)}{\partial b}=\frac{n_{1}}{b}+\sum_{i=1}^{n_ {1}}\ln t_{i}-(a^{*}+1)\sum_{i=1}^{n_{1}}\frac{t_{i}^{b}\ln t_{i}}{1+t_{i}^{b }}+a^{*}K_{2}=0, \tag{12}\]  
where  
\[K_{1}=\sum_{i=n_{1}+1}^{n_{1}+n_{2}}\frac{(1+t_{i}^{b})^{-a^{*}}\ln{(1+t_{i}^{ b})}-(1+t_{i}^{b})^{-a^{*}}\ln{(1+t_{i}^{b})}}{(1+t_{i}^{b})^{-a^{*}}-(1+t_{i}^{b })^{-a^{*}}},\]  
and  
\[K_{2}=\sum_{i=n_{1}+1}^{n_{1}+n_{2}}\frac{r_{i}^{b}(1+r_{i}^{b})^{-(a^{*}+1)} \ln r_{i}-l_{i}^{b}(1+l_{i}^{b})^{-(a^{*}+1)}\ln l_{i}}{(1+l_{i}^{b})^{-a^{*}} -(1+t_{i}^{b})^{-a^{*}}}.\]  
For \(j=1,2,\ldots,s\), we determine, respectively, the MLEs of \(a_{j}\) and \(a^{*}\) as \(\hat{a}_{j}\) and \(\hat{a}^{*}\), and the MLE of \(b\) is expressed as \(\hat{b}\). Based on (11), we can derive that \(\frac{m_{1}}{a_{1}}=\frac{m_{2}}{a_{2}}=\cdots=\frac{m_{s}}{a_{s}}\). Thus,  
\[\frac{\hat{a}_{j}}{m_{j}}=\frac{\hat{a}^{*}}{n},j=1,2,\ldots,s. \tag{13}\]  
Before obtaining the solution through (11)-(13), the uniqueness and existence of solutions to the likelihood equations require verification. Theorem 1: _For competing risks and middle-censored data following the Burr-XII distribution, when parameter \(b\) is known, the MLEs of parameters \(a_{1},a_{2},\ldots,a_{s}\) are always existent and unique._Proof. : Through (11)-(13), the likelihood equations are equivalent to the following:  
\[