the Gaussian kernel density method as below: $$\label{initial-appro}
p_{a,n}(z,0):=\frac{1}{n}\sum_{i=1}^{n}G_{n^{-1/4}}\left(z-g\left(z_{t_i},t_i,t_i\mid a\right)\right)$$ where $G_h$ denote the Gaussian kernel function with kernel width $h$. In this paper, we simply select the kernel width as $\frac{1}{n^4}$ as can guarantee the function converges to $p_{a}(.,0)$ in the $L^1$ norm for all $a$ \cite{}.

For the baseline hazard $\lambda$, without loss of generality, we set $t_{0}=0$, $\theta_{0}=1$ and let $0\leq t_{1}<t_{2}<\cdots<t_{n}<\infty$ be the ordered statistics of the $n$ observed failure time. A step-wise version of the non-parametric baseline hazard is constructed as below: $$\label{step-wise-lambda}
\lambda_{n}^{s}\left(t\right):=\sum_{i=1}^{n}\theta_{i}\cdot I_{[t_{i-1},t_{i})}\left(t\right)$$ where $\theta=\left\{ \theta_{i}\geq0:i=1,\dots,n\right\} $ is a set of parameters to be estimated. For each profile of the parameters $\Omega_{n}=\left(\lambda_{n}^{s},a,b\right)$, we can define the log likelihood as below: $$\label{likelihood1}
l_n\left(a,b,\lambda_{n}^{s}\right):=\frac{1}{n}\sum_{i=1}^{n}\left(
\begin{aligned}
&\log p_{a,n}\left(g\left(z_{t_i},t_i,t_i\mid a\right),0\right)
+\log \mathcal{J}_{z_{t_i}|a}\left(t_i\right)+\log\lambda_{n}^{s}\left(t_{i}\right)\\
&+b^{T}z_{t_{i}}-\int_{0}^{t_{i}}\lambda_{n}^{s}\left(t_{i}-\tau\right)\exp\left(b^{T}g\left(z_{t_{i}},t_{i},\tau|a\right)\right)d\tau
\end{aligned}
\right).$$ The estimator resulting from maximizing is denoted as $\hat{\lambda}^{s}$ and $\hat{b}$ and $\hat{a}$.

\[remark1\] The first order condition of the optimization problem indicates the relation $$\label{constraint-for-lambda}
\theta_{i}=\frac{1}{\sum_{j=i+1}^{n}\int_{0}^{t_{i}-t_{i-1}}\exp\left(b^{T}g\left(z_{t_{j}},
t_{j},t_{j}-t_{i}+\tau|a\right)\right)d\tau}$$ at the optimal point $\hat{\lambda}_{n}^{s}$ and $\hat{b}$ and $\hat{a}$. Relation can be inserted as a set of constraints back into the optimization problem , which helps sharply reduce the dimension of the original problem.

Variable Selection  {#variable selection}
-------------------

A penalty function can be 