I am looking at the following post in the Statistics Stack Exchange:
where there was a question about the hazard function in survival analysis and intensity function in Poisson process. The answer was superb and in the end the responder had written (and I rephrase):
Now, we can extend the paradigm to say that the failed system is instantaneously replaced by a brand-new system that begins operating at time $T$, but the analysis now begins anew and the hazard rate $\hat{h}(t)$ for the replacement is $h(t−T)$ etc. In other words, the probability that the replacement is struck dead in $(T,T+\Delta t]$ is $h(0)Δt$, not $h(T)\Delta t$.
Let us hypothetically assume we are extending the survival process and magically the system gets revived. In that case, it appears to me that we need a new hazard rate $\hat{h}(t)$ because we assume independence of events in Poisson process? Suppose we have conditional hazard function instead of pure hazard function so the hazard rate is dependent on the history, then will the natural extension of survival analysis to Poisson process will be a Hawkes process? And how do we prove the new hazard function will be $h(t-T)$?