the parameters indexing model (1) can be obtained by solving the estimating equation:

$$\begin{aligned}
\bigintsss_0^{\infty} \sum_{i, a, a^*}&\left\{\begin{pmatrix} a \\ a^* \end{pmatrix} - \frac{\sum_{a,a^*}\hat E\left[\begin{pmatrix} a \\ a^* \end{pmatrix}
R_i^j(t)W_i(\lfloor t \rfloor ,a,a^*)e^{\alpha_{1j}a + \alpha_{2j}a^*}\right]}
{\sum_{a,a^*} \hat E\left[R_i^j(t)W_i(\lfloor t \rfloor ,a,a^*)e^{\alpha_{1j}a + \alpha_{2j}a^*}\right]}\right\}\cdot \\
& \cdot R_i^j(t)W_i(\lfloor t \rfloor ,a,a^*)\left(dN_i^j(t) - \lambda_0^j(t)e^{\alpha_{1j}a + \alpha_{2j}a^*}\right) = 0\end{aligned}$$ where $R_i^j(t) = I(T_i\ge t, D_i = j)$; $dN^j_i(t) = I(T_i = t, D_i = j)$ and $I(.)$ denotes the indicator function. Besides, $$\begin{aligned}
W_i(\lfloor t \rfloor ,a,a^*) = \frac{\prod_{s:t_s\le \lfloor t \rfloor } {\text{Pr}}(M_{s,i}|A_i = a^*, \overline{M}_{s - 1,i}, \overline{L}_{s,i}, T_i \ge t_s)}{\prod_{s:t_s\le \lfloor t \rfloor} {\text{Pr}}(M_{s,i}|A_i = a, \overline{M}_{s - 1,i}, \overline{L}_{s,i}, T_i \ge t_s)}\times \frac{I(A_i = a)}{{\text{Pr}}(A_i=a|L_{0,i})}\end{aligned}$$ denotes the weight of individual $i$ at time $t$ and at exposure levels $a$ and $a^*$. The second component of the weight ensures that the exposure-outcome association is adjusted for confounding by $L_0$. It creates a pseudo-population in which the exposure is no longer associated with $L_0$ and hence removes confounding by $L_0$ [@lange12]. The first component of the weight then distinguishes between the direct and indirect paths by correcting for the fact that the observed mediator value at each time point before $t$ 