# Low intensity Poisson estimation

I have a collection of Poisson processes each with an unknown $\lambda$.

I would like to estimate $\lambda$ for each process.

for each process I could take either the total number of event over the total time or the inverse of the mean of the waiting times.

Given that the dataset is quite small, with many of the processes having less than 5 total events, is is better to use the estimate based on the waiting times or the total count?

The replies to this question suggest that the count over the time is the best estimator, but no mention of the waiting times.

• The counter-question is how can you know that you have Poisson processes? It seems more likely that you have count data and your first and/or simplest guess at a generating process is a Poisson. Why not use both methods and see if they agree closely? If they do, you're fine; if they don't, your assumption seems dubious. But whatever you do, no white magic makes up for small sample sizes. – Nick Cox Aug 31 '13 at 8:35
• I know that the individual processes are Poisson because the superposition of the processes (which were collected at the same time) is well described by Poisson process – dylan2106 Aug 31 '13 at 17:07
• This is not a valid conclusion in general. While it is true that the the superposition of independent Poisson processes is also a Poisson process, it is not necessarily true that the component processes of a Poisson process are Poisson. You should analyze each individual process if you can. – soakley Aug 31 '13 at 17:47
• could you give an example of a non-Poisson process that would combine to give a poisson process. – dylan2106 Aug 31 '13 at 21:43
• I can give you a reference. Look up the Palm-Khintchine theorem on Wikipedia. – soakley Sep 1 '13 at 12:52

## 1 Answer

Use the total number of events over the total time. If your last event does not occur at the end of the time window, then you have data that you should use, but will be throwing away with the other method. That data is a length of time, and it should be included in your denominator. Censored data is valuable; use it.

Total number of events over total time is an unbiased estimator for the rate. The expected number of events in a fixed time $T$ is $\lambda T.$ So if we call this estimator $\hat \lambda,$ then $$E[\hat \lambda ]= {{\lambda T} \over {T} }= \lambda .$$

• yes I see how using the wait times would be ignoring the censored data at the end... so it is a biased(?) estimator compared to the count. However it still seems like the wait times will give me a better idea of an underlying distribution of λs since it can take continuous values, unlike the count which is discrete – dylan2106 Aug 31 '13 at 17:15
• I agree that it is a good idea to look at the distribution of the waiting times. A Lilliefors test against an exponential could help to confirm your process is Poisson. But if you do have data with censored times at the end, I still think the approach above is the best way to estimate the $\lambda$'s. – soakley Aug 31 '13 at 17:51