\begin{equation}\label{loglikelihood}
\log \mathcal{L}_\ell (\theta^{(\ell)})=\sum_{i=1}^{n_\ell} \log(S^{(\ell  )}(\tilde{T}_i^{(\ell)}))+ \sum_{i=1}^{n_\ell}\sum_{j=1}^k I\{X^{(\ell)}(T_i^{(\ell)})=j\}\log(\alpha^{(\ell)}_{0j}(T_i^{(\ell)},\theta^{(\ell)}_{0j})).
\end{equation}
