Assume I have a process that I assume to be Poisson, meaning that the waiting time between events is exponentially distributed. As a concrete example, assume I want to estimate the waiting time to a car accident at an intersection. These accidents happen infrequently, so the data I have must be used efficiently. It takes $X_1$ days for the first accident to occur, $X_2$ days after the first accident for the second, etc. It's been $X_C$ days (and counting) since the last accident.
What's the best (unbiased, minimum variance) way to estimate the waiting time in this situation, where it's been some time since the last Poisson event occurred and I want to take advantage of my knowledge of $X_C$?
Edit: I am interested in the exponential rate parameter. The interesting part of the question is including the censored value $X_C$ in the estimate. The terminology for this somehow escaped me when I wrote the original question.