I am interested in a situation when the time to event is observed partially. In the right-censoring situation, we know the start time of the disease $s$ but do not observe the death time $d$, because of censoring time $c_r$. So time interval $c_r-s$ is observed.
What situation am I in, when the death is observed at time $d$, but the start time of the disease $s$ is not known due to censoring time $c_l$? So the time interval $d-c_l$ is observed.
For the right-censoring the contribution to the likelihood is $S_\theta(c_r-s)$, where $S_\theta(t)$ is a parametric survival function. What would be the contribution to the likelihood in the second situation: is it $1-S_\theta(d-c_l)$?
Consider an example of consumer buying a product. Suppose we have data from time $c_l$ to $c_r$ about repeated purchases and we model inter-purchase time. Three cases are possible:
- The time of the current purchase is observed but the time of the next purchase is after $c_r$ (not observed) - right-censoring.
- The time of the previous purchase is before $c_l$ (not observed) but the time of the current purchase is observed - the situation under question. I could just ignore this observation, but it might be informative (especially in a situation with rare purchases).
- The interval between the two purchases falls into $(c_l,c_r)$ - full observation.