based on iterative calculation of the values of the Bayesian criterion and the likelihood function. Instead of using the iterative procedure, we simply set $\Lambda_n=n^{-\frac{1}{4}}$ in order to reduce the computation load.

The original adaptive LASSO method is designed for the OLS procedure ([@zou2006adaptive]), but it turns out this method applies very well to the likelihood-based estimation procedures as ours. The oracle property of the adaptive LASSO estimator, $\hat{b}_{AL}$, will be proved in Appendix \[theorem-ALASSO\].

LARGE SAMPLE PROPERTIES
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The consistency and asymptotic normality of the estimators, $\hat{a}$, $\hat{b}$ and $\hat{\lambda}_{n}^{s}$, constructed in section 2.1 will be established in this section. We will also show the oracle property of the adaptive LASSO estimator $\hat{b}_{AL}$ and the consistency and asymptotic normality of $\hat{\lambda}_{AL}^{s}$ and $\hat{a}_{AL}$.

Large Sample Properties of $\hat{a}$, $\hat{b}$ and $\hat{\lambda}^{s}$ {#large sample property}
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Let $\Omega:=\left(a,b,\lambda\right)$ be a given profile of parameter and in particular, $\Omega_{0}$ be the true parameter. Define $$\label{func V}
{\scalebox{0.8}{$
V_{\Omega}:=p_{a}\left(g\left(Z_{T},T,s\mid a\right),0\right)\mathcal{J}_{Z_T|a}(T)\exp\left(-\int_{0}^{T}\lambda\left(T-s\right)\exp\left(b^{T}g\left(Z_{T},T,s\mid a\right)\right)ds\right)\lambda\left(T\right)
\exp\left(b^{T}Z_{T}\right)
$}}$$ as a random variable with $Z_{T}$ and $T$ being the random variables following the joint pdf associated with $\Omega_{0}$, denoted as $pdf_{\Omega_0}$. The following technical conditions are needed for the consistency result:

$C1$. The domain $A$ and $B$ are 