using the function $q$ and $\lambda$:

$$\label{joint-pdf}
{\scalebox{0.8}{$
pdf\left(z,t\mid a_{0},b_{0},\lambda_{0}\right)=\tilde{p}\left(z,t\right)\times\exp\left(-\int_{0}^{t}\lambda_{0}\left(t-s\right)\exp\left(b_{0}^{T}g\left(z,t,s\right)\right)ds
\right)\times\lambda_{0}\left(t\right)\exp
\left(b_{0}^{T}z\right)$}}$$

where the function $\tilde{p}$ is the time-dependent pdf induced by the longitudinal process $\left\{ Z_{t}\right\} $ and by [@XQZhang2017] it can be expressed as $$\label{p-tilde}
\tilde{p}(z,t)=p\left(g\left(z,t,t\right),0\right)\cdot \mathcal{J}_{z,t},$$ The function $p(,.0)$ is the initial pdf induced by $Z_0$ and for every $t$, $\mathcal{J}_{z}(t)$ denotes the Jacobian of the function $g\left(.,t,t\right)$ evaluated at the point $z$. The function $g$ is solely determined by $q$ through solving a family of initial value problems (IVPs). Namely for every fixed $z$ and $t$, $g\left(z,t,.\right)$ is the solution to the following ordinary differential equation (ODE) for $s\in\left(0,t\right)$: $$\label{ode}
z'\left(s\right) = -q\left(z\left(s\right),t-s\mid a_{0}\right)$$ subject to the initial condition $g\left(z,t,0\right)=z$.

In addition to estimating model parameters, in practice it is also important to select the significant covariates. Variable selection approaches has been extensively studied by many authors. The least absolute shrinkage and selection operator (LASSO) was presented by [@tibshirani1996regression]. [@fan2001variable] developed the nonconcave penalized approach (SCAD) for variable selection which applies to likelihood-based estimation procedures, including the MPL procedure for the Cox model ([@fan2002variable]). [@zou2006adaptive] developed an Adaptive LASSO approach and showed its oracle property under a general set of conditions. The estimation procedure proposed in 