support $\mathbb{R}^{p}\times \mathbb{R}_{+}$, and the second-step optimization is equivalent to the optimization of a conditional log-likelihood function formed by the sum of $\log\rho\left(z_{t_{m_i}},t_{m_i}\right)$ for all $m_i$’s, the full support condition in $C6$ can be relaxed.

In addition, from the proof of the theorem \[theorem-identification\], it is clear that the main difficulty to achieve the injectivity of the map from the parameter space to the space of all joint pdfs lies in the exclusion of the possibility that different $a$ could lead to the same joint pdf, which is exactly the condition $C5$ and $C6$ designed for. In contrast, when $a_0$ is given, the identification of $b_0$ and $\lambda_0$ becomes trivial and doesn’t require any further conditions like $C5$ and $C6$. So when the estimation of $a_0$ can be separated out, $C5$ and $C6$ are redundant.

Proof of the validity of the two-step procedure is in Appendix \[proof-two-step\]. A latent assumption behind the proof is that the observational time is uninformative and the total number of observations, $m_i$, is at least $2$ for all subjects. Unlike the joint model discussed in [@tsiatis2004joint], we don’t have to assume $m_i$ greater than the dimension of the covariates. This fact makes our two-step procedure 