naturally incorporated into the estimation and determine the non-zero component of the coefficients $b$ automatically. There exist many types of penalty functions in the literature, such as LASSO ([@tibshirani1996regression]), adaptive LASSO ([@zou2006adaptive]), SCAD ([@fan2001variable]), MCP. For convenience of calculation, we choose the adaptive LASSO penalty function defined by:

$$\label{penality-funcion}
P_{\Lambda}\left(b_{j}\right):=\Lambda\frac{\left|b_{j}\right|}{\left|\hat{b}_{j}\right|^{r}}$$

where $b_{j}$ is the $j$-th component of the the vector $b$, $\hat{b}_{j}$ is a consistent estimator of $b_{j}$ (say the estimator generated by maximizing Eq. ), $\Lambda$ is a tuning parameter and $r>0$ (for simplicity of calculation, we let $r=2$ throughout the paper). The penalty function is added into the likelihood function and forms the following maximization problem : $$\label{likelihood4}
\max_{a,u,\lambda^{s}}\left(l_{n}\left(a,\hat{b}+u,\lambda^{s}\right)+\sum_{j=1}^{p}P_{\Lambda_{n}}
\left(\hat{b}_{j}+\frac{u}{\sqrt{n}}\right)\right)$$ where the choice of tuning parameter $\Lambda_{n}$ depends on the sample size $n$. Denote $\hat{a}_{AL}$, $\hat{\lambda}_{AL}^{s}$ and $\hat{b}_{AL}$ ($:=\hat{b}+\hat{u}_{AL}$ with $\hat{u}_{AL}$ being the solution to ) as the estimator corresponding to maximize Eq. .

In the original work ([@zou2006adaptive]), a wide range of candidate values can be selected for the tuning parameter $\Lambda_n$ as long as the asymptotic property $\Lambda_n\cdot n^{-\frac{1}{2}}\rightarrow 0$ and $\Lambda_n\cdot n^{\frac{1}{2}}\rightarrow \infty$ hold. In practice, there are multiple ways to select the value of $\Lambda_n$ such as the Bayesian criterion method \cite{} which is 