## 3 Frequentist Estimation
### Maximum Likelihood Estimators
\begin{cases}d_{1}(a^{*},b)=\frac{n_{1}}{a^{*}}-\sum_{i=1}^{n_{1}}\ln{(1+t_{i}^{b} )}+\sum_{i=n_{1}+1}^{n_{1}+n_{2}}\frac{\Psi_{\gamma_{i}^{-a^{*}}}\ln{\psi_{ \gamma_{i}}}-(\psi_{\gamma_{i}})^{-a^{*}}\ln{\psi_{\gamma_{i}}}}{\psi_{\gamma_{ i}^{-a^{*}}}-\psi_{\gamma_{i}^{-a^{*}}}}=0\\ d_{2}(a^{*},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^{*}\sum_{i=n_{1}+1}^{n_{1}+n_{2}} \frac{t_{i}^{b}\Psi_{\gamma_{i}}^{-a^{*}}}{\psi_{\gamma_{i}}^{-a^{*}}-\psi_{ \gamma_{i}^{-a^{*}}}}=0\end{cases} , \tag{14}\]  
\[\text{where }\psi_{\gamma_{i}}=1+t_{i}^{b}\text{ and }\psi_{\gamma_{i}}=1+t_{i}^{b} \text{. Therefore,} \tag{15}\]  
\[\frac{\partial d_{1}(a^{*},b)}{\partial a^{*}}=-\frac{n_{1}}{a^{*2}}-B, \tag{16}\]  
where  
\[B=\sum_{i=n_{1}+1}^{n_{1}+n_{2}}\frac{\psi_{\gamma_{i}}^{-a^{*}}\psi_{\gamma_{ i}^{-a^{*}}}(\ln{\psi_{\gamma_{i}}}-\ln{\psi_{\gamma_{i}}})^{2}}{(\psi_{\gamma_{ i}}^{-a^{*}}-\psi_{\gamma_{i}^{-a^{*}}})^{2}}.\]  
When \(b\) is a known constant, we just obtain the solution of \(d_{1}(a^{*},b)=0\) to obtain \(\hat{a}^{*}\). It is obvious that \(\frac{\partial d_{1}(a^{*},b)}{\partial a^{*}}<0\), and that  
\[ \begin{cases}\lim\limits_{a^{*}\to 0}d_{1}(a^{*},b)=+\infty\\ \lim\limits_{a^{*}\rightarrow+\infty}d_{1}(a^{*},b)=-\sum_{i=1}^{n_{1}}\ln{(1+ t_{i}^{b})}-\sum_{i=n_{1}+1}^{n_{1}+n_{2}}\ln{\psi_{\gamma_{i}}}<0\end{cases} , \tag{17}\]  
so \(d_{1}(a^{*},b)\) is monotonically decreasing with respect to \(a^{*}\), and takes both positive and negative values, which indicates the existence and uniqueness of the zero point \(\hat{a}^{*}\). Therefore, through (13), we can conclude that when \(b\) is fixed, for parameters \(a_{1},a_{2},\ldots,a_{s}\), their MLEs always exist and are unique and finite. Theorem 2.: _For competing risks and middle-censored data following the Burr-XII distribution, in the case that \(a_{1},a_{2},\ldots,a_{s}\) and \(b\) are all unknown, when there exists \(i\) such that \(t_{i}<1\), the MLEs \(d_{1},d_{2},\ldots,d_{2}\) and \(\hat{b}\) all exist and are unique._  
Proof. : According to (9), the joint likelihood function is expressed as:  
\[L(\textbf{a},b)\propto\prod_{j=1}^{s}\prod_{i=1}^{n_{1}}(f(t_{i},j;\textbf{a}, b))^{I(D_{i}=j)}\prod_{j=1}^{s}\prod_{i=n_{1}+1}^{n_{1}+n_{2}}(F(r_{i},j; \textbf{a},b)-F(l_{i},j;\textbf{a},b))^{I(D_{i}=j)},\]  
in accordance with the Lagrange mean-value theorem, there is a \(\xi_{i}\in[L_{i},R_{i}]\) satisfying \([F(r_{i},j;\textbf{a},b)-F(l_{i},j;\textbf{a},b)]=(R_{i}-L_{i})f(\xi_{i},j; \textbf{a},b)\). So we just need to prove that when the data are uncensored, the MLEs of the parameters \(a^{*}\) and \(b\) are existent and unique. Through (24), and for \(i>n_{1}\), for convenience, we substitute \(t_{i}^{*}\) with \(t_{i}\), then we can obtain that  
\[\hat{a}^{*}=n/\sum_{i=1}^{n}\ln{(1+t_{i}^{b})}. \tag{18}\]  
Then, \(\hat{b}\) is the solution of the equation given as follows:  
\[z(b)=\frac{n}{b}+\sum_{i=1}^{n}ln{t_{i}}-(\frac{n}{\sum_{i=1}^{n}\ln{(1+t_{i}^ {b})}}+1)\sum_{i=1}^{n}\frac{t_{i}^{b}ln{t_{i}}}{1+t_{i}^{b}}=0. \tag{19}\]  
We can derive the derivative of \(z(b)\) as follows:  
\[z^{{}^{\prime}}(b)=-\frac{n}{b^{2}}-\frac{n}{(\sum_{i=1}^{n}\ln{(1+t_{i}^{b})} )^{2}}(\sum_{i=1}^{n}\frac{t_{i}^{b}ln{t_{i}}}{1+t_{i}^{b}})^{2}-\frac{n}{\sum_{ i=1}^{n}\ln{(1+t_{i}^{b})}}\sum_{i=1}^{n}\frac{t_{i}^{b}(\ln{t_{i}})^{2}}{(1+t_{i}^{b}) ^{2}}<0, \tag{20}\]and it is obvious that  
\[ \begin{cases}\lim\limits_{b\to 0}z(b)=+\infty\\ \lim\limits_{b\to+\infty}z(b)=\lim\limits_{b\to+\infty}z_{1}(b)+\lim\limits_{b \to+\infty}z_{2}(b),\end{cases} \]  
where \(z_{1}(b)=-\frac{n}{\sum_{i=1}^{n}\ln\left(1+t_{i}^{b}\right)}\sum_{i=1}^{n} \frac{t_{i}^{b}Int_{i}}{1+t_{i}^{b}}\) and \(z_{2}(b)=\sum_{i=1}^{n}\frac{\ln t_{i}}{1+t_{i}^{b}}\).