as the set of indices with $b_{0,j}\not=0$ for $j\in\mathcal{A}$ and $\hat{\mathcal{A}}\subset\left\{ 1,\dots,p\right\} $ as the set of indices with $\hat{b}_{0,j}\not=0$ for $j\in\hat{\mathcal{A}}$, then $\hat{b}_{j}\rightarrow_{p}b_{0,j}$ for all $j\in\mathcal{A}$ and $Prob\left(\hat{\mathcal{A}}=\mathcal{A}\right)\rightarrow1$;

(2). denote $I_{\mathcal{A}}:=\left\{ E_{\Omega_{0}}\left(\nabla_{b_{i}}\log\left(V_{\Omega}\right)\nabla_{b_{j}}\log\left(V_{\Omega}\right)\right)\right\} _{i,j\in\mathcal{A}}$, $\hat{b}_{L,\mathcal{A}}=\left(\hat{b}_{L,j}\right)_{j\in\mathcal{A}}$ and $b_{0,\mathcal{A}}=\left(b_{0,j}\right)_{j\in\mathcal{A}}$, $\sqrt{n}\left(\hat{b}_{L,\mathcal{A}}-b_{0,\mathcal{A}}\right)\rightarrow_{d}N\left(0,I_{\mathcal{A}}^{-1}\right)$ .

Extension
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When the longitudinal observations are available at the observation time before failure occurs at $T$, i.e. the input data has the form $\left\{\left(Z_{i,j},t_{i,j}\right):\,j=1,\dots,m_i;\,i=1,\dots,n\right\}$ with $m_{i}>1$ for $i=1,\dots,n$. A two-step procedure can be applied to estimate the parameter $\Omega_{0}$, and the resulting estimator turns out to be consistent and have asymptotically normal distribution even without the assumption $C5$ and $C6$. Instead, the following Markovian-style condition are required: $$\label{markovian}
E\left(\epsilon(t-s)\mid Z(t),T\geq t\right)\equiv E\left(\epsilon(t-s)\mid Z(t)\right)$$ where $\epsilon(t)$ is the instantaneous variational rate of the longitudinal process as specified in model model . Eq. implies that the conditional mean trajectory that reaches a given realization, $Z(t)$, at the observational time $t$ won’t be affected by whether or not the event has already occurred. Formally, the two-step algorithm is stated as following:

Step 1: estimate the parameter $a$ through minimizing the empirical mean of the $L^{2}$ distance between the empirical longitudinal trajectories observed for each individual $i$ and the theoretical mean trajectories passing through 