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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.

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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? –  cardinal Mar 30 '12 at 1:33
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1 Answer

up vote 6 down vote accepted

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 $E[\tfrac{N}{t}|\lambda] = \tfrac{E[N|\lambda]}{t} = \tfrac{\lambda t}{t} = \lambda$.

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Maybe this is a language issue of mine or I'm just misreading your answer, but did you mean "not obviously" or "obviously not"? –  cardinal Mar 30 '12 at 20:30
@cardinal: my wording was deliberate, but feel free to read it your way: "obviously not" is a subset of "not obviously" and I am prone to understatement to make a point. –  Henry Mar 30 '12 at 20:38
Ok, perhaps I am tired or this is some regional usage difference? I would interpret the implications of the statements "A bat is not obviously a bird" and "A bat is obviously not a bird" as very different, for example. –  cardinal Mar 30 '12 at 20:44
Great answer. Wow, that's obvious in hindsight. –  dsimcha Mar 30 '12 at 20:45
(+1) By the way. –  cardinal Mar 30 '12 at 20:53
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