# Estimating Poisson waiting times: using the last, censored sample?

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.

• This question isn't quite clear. Are you interested in estimating the exponential rate parameter $\lambda$ and incorporating the final censored observation in that estimate? Mar 30, 2012 at 1:33

Suppose there have been $N$ accidents in time $t = X_C +\sum_{i=1}^N X_i$. Since $t$ is the time from when measurements started to now, it is not obviously a random variable, though it incorporates $X_C$ and all the other time measurements.
Then an obvious estimator for the rate, particularly given a memoryless process, is $\hat{\lambda}=\tfrac{N}{t}$. This is unbiased in the sense that $E[\tfrac{N}{t}|\lambda] = \tfrac{E[N|\lambda]}{t} = \tfrac{\lambda t}{t} = \lambda$.