Probability model for the time to event when the origin is not known 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.


If you are modeling the inter-purchase time as explained at the end of your question, with time = 0 for the survival origin reset at each purchase, then the inter-purchase time would seem to be right censored even in Case 2 (the first purchase after study start).
A right-censored observation is one for which you know a lower limit for the time between time = 0 and the event. The definition of time = 0 thus matters a lot.
If time = 0 is defined as the time of the previous purchase (and thus is reset at each purchase), then you do have a lower limit to the elapsed inter-purchase time in Case 2. You know that the prior purchase (if any) was prior to study start, so the corresponding prior inter-purchase time was at least the time elapsed since study start. In terms of inter-purchase time that's as right censored as the time between the last purchase and the end of the study (Case 1). The value of what you are modeling, the inter-purchase time, has a lower limit but no upper limit, the definition of right censored.
A few warnings that come quickly to mind: This would not necessarily hold for other analysis approaches that use a different definition of time = 0. You might have problems with modeling actual customers who made no purchases during the study period; the length of the study period could set a truncation limit. Complications would arise if you were incorporating covariate values at time = 0 into your model and didn't know the covariate values for purchases that happened prior to study start.
