I am faced with a basic question about counting process and its intensity process used in survival analysis. It is actually the textbook example from Aalen's Survival and Event history analysis book.
Some notations first
"First we generate 10 survival times, $T_i$, from a Weibull distribution with hazard rate $\alpha(t)=0.0031t^3$ and 10 censoring times, $C_i$, from a uniform distribution over the interval (0,10) too obtain 10 censored survival times $\widetilde{T}_i=\text{min}(T_i, C_i)$ and censoring indicators $D_i=I(\widetilde{T}=T_i)$. "
The intensity process $\lambda(t)$ is defined as summation of all individuals' intensity process, and it turns out to be the summation of the product of hazard rate function $\alpha_i(t)$ and at risk process $Y_i(t)$. \begin{align} \lambda(t) &=\sum^n_{i=1} \lambda_i(t) \nonumber \\ &=\sum^n_{i=1} \alpha_i(t)Y_i(t) \end{align}
$i$ is $i$th observation in the data. We have n independent observations
My question is how to compute this the cumulative intensity process $\Lambda(u)=\int_0^t \lambda(u) du $?
Below are some of my thoughts,
\begin{align} \Lambda (t) &=\int_0^t \lambda(u) du \nonumber \\ &=\int_0^t \sum^n_{i=1} \alpha_i(u)Y_i(u) du \nonumber \\ &=\sum^n_{i=1}\int_0^t \alpha_i(u)Y_i(u) du \end{align}
We know at risk process $Y_i(t)$ is an indicator function
$$ Y_i(t) = \left\{ \begin{array}{rl} 1 &\mbox{ if $\widetilde{T}_i \ge t$} \\ 0 &\mbox{ if $\widetilde{T}_i < t$} \end{array} \right. $$
Now the question is how to compute $\int_0^t \alpha_i(u)Y_i(u) du$ ?
If $\widetilde{T}_i \ge t$, $\int_0^t \alpha_i(u)Y_i(u) du$ can be reduced to $\int_0^t \alpha_i(u)du$. If $\widetilde{T}_i$ is between 0 and t, there are two scenarios. For $u \leq \widetilde{T}_i$, $Y_i(u)=1$ . For $\widetilde{T}_i \leq u$, $Y_i(u)=0$ . Thus, $\Lambda(t)$ can be expressed as:
$$ \Lambda(t) = \left\{ \begin{array}{rl} \sum^n_{i=1}\int_0^t \alpha_i(u) du &\mbox{ if $\widetilde{T}_i \ge t$} \\ \sum^n_{i=1}\int_0^{\widetilde{T}_i} \alpha_i(u)du &\mbox{ if $\widetilde{T}_i < t$} \end{array} \right. $$
Do I understand the computation of $\Lambda(t)$ correctly? thanks,