the point $\left(Z_{t_{i,m_{i}}},t_{i,m_i}\right)$:

$$\label{mean-L2-distance}
\min_{a\in A}\frac{1}{n}\sum_{i=1}^{n}\frac{1}{m_i}\sum_{j=1}^{m_{i}}\left(g\left(z_{t_{i,m_{i}}},t_{i,m_{i}},t_{i,m_{i}}-t,_{i,j}\mid a\right)-z_{t_{i,j}}\right)^{2}.$$

It turns out when $n\rightarrow\infty$, the estimator $\hat{a}_{E}$ of solving the problem is consistent.

Step 2: replace $a$ by $\hat{a}_{E}$ and maximize the likelihood function or for the parameter $b$ and $\lambda$.

It turns out that the resulting estimators do have the same properties as stated in Theorem \[theorem-consistency\] or \[theorem-ALASSO\]. The two-step procedure separates the estimation of $a_{0}$ from the estimation of $b_{0}$ and $\lambda_{0}$. Thanks to this separation, in the second step, the initial pdf and the Jacobian can be completely removed from the likelihood function or because they only depends the parameter $a$ and the fixed underlying true distribution. In the other words, once if the parameter $a$ is replaced by its first-step estimator $\hat{a}_E$, the component of $\tilde{p}$ becomes constants, and can be deleted from the second-step maximization problem without any impact on the final estimators. As a consequence, there is no need to calculate the Jacobian $\mathcal{J}_{z_{t_{m_i}}|a}\left(t_{i,m_i}\right)$, which makes the two-step procedure running much faster than the original procedure because the computation of the Jacobian is the most time-consuming part.

Due to the fact that the conditional density function $$\rho(z,t):=\exp\left(-\int_{0}^{t}\lambda\left(t-s\right)
\exp\left(b^{T}g\left(z,t,s\mid a\right)\right)ds\right)\lambda\left(t\right)
\exp\left(b^{T}z\right)$$ has already had the full 