briefly discuss the adjustment of our procedure to deal with the case where more longitudinal observations are available.

Likelihood Function {#likelihood function}
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Define $A\subset\mathbb{R}^{d}$ as the domain of all possible values of the parameter $a$, $a_{0}\in A$ as the true parameter. Similarly, Define $B\subset\mathbb{R}^{p}$ as the domain of $b$, $b_{0}\in B$ as the true parameter. For every fixed $a\in A$, define $g\left(.\mid a\right)$ as the solution trajectories to the IVPs conditional on $a$. When the analytic form of $g\left(.\mid a\right)$ is not available, we can use its numerical approximation in place. There are many efficient numerical solvers to the IVPs . In this paper, we pick up the Euler’s method \cite{} for the purpose of being simple and illustrative. Similarly, write $\mathcal{J}_{z|a}(t)$ as the Jacobian of $g\left(.,t,t\mid a\right)$ and when necessary it can be replaced by its numerical version.

By [@XQZhang2017], the initial pdf $p(.,0)$ is uniquely determined by the function and the joint pdf . In particularly, given the joint pdf , there is a well defined map $a\mapsto p_{a}(.,0)$ from the parameter space $A$ to the space of the pdfs over $\mathbb{R}^{p}$. Therefore, given the input data and a fixed parameter $a$, we can estimate $p_a(.,0)$ by 