(2) the event process $Y$ which determines the time to event and its conditional expectation has the form , i.e. $$\label{conditonal-hazard}
\lambda\left(t\mid z\right)=
\lambda_{0}\left(t\right)\exp\left(b_{0}^{T}z\right):=
E\left(-dY(t)\mid Y(t^{-})=1, Z(t^{-})=z\right).$$ It turns out by [@XQZhang2017], combining and yields a complete specification of the joint model in the sense that if two joint models share a common pair of the conditional hazard function and the conditional expectation , all the distributions in interest arising from the two joint models are identical. The equation is the key to derive an explicit expression of the joint probability density function (pdf) of $Z_T$ and $T$ which helps design our estimation procedure. To our best knowledge, there has not been any previous works attempting to specify the longitudinal process as . We hope our work could provide some hints to the future development of this field.

In model and , there are three sets of parameters, $a_{0}$, $b_{0}$ and $\lambda_{0}$. Among them, $\lambda_{0}$ has infinite dimension. Our procedure will estimate the three kinds of parameters through maximizing the full information likelihood function, where the likelihood is constructed from the joint probability density function (pdf) of $Z_T$ and $T$. By [@XQZhang2017], this joint pdf is expressed as below by 