# Test hypothesis point process is Poisson [duplicate]

I have some data and I would like to test the hypothesis that they come from a homogeneous Poisson process. I can of course look at the inter event times and test if these are exponentially distributed. However this misses lots of reasons why it might not be Poisson it seems . Is there a list of tests, or a particular good test, that I can use that does more than look at the set of inter event times between consecutive events?

If you take all the differences between arrival times, that is not just consecutive arrival times, can this be used to make a more powerful test for example?

## marked as duplicate by gung♦, whuber♦Mar 15 '15 at 18:51

• Is there some way to apply the Kolmogorov-Smirnov test here other than looking at inter arrival times between consecutive arrivals? If I look at all inter arrival times (that is all roughly $n^2$ of them for $n$ points), then I am not sure how to apply KS any more. – felix Aug 6 '14 at 3:12
• It seems like you are looking for a free lunch. You might consider the distribution of the arrival times for alternate events (i.e. what is the probability that one event has occurred). I believe this just $\frac{e^{\lambda t}(\lambda t)^1}{1!}$ You still have $n$ intervals to compare. But I don't think you can then simultaneously try to test P(0) (first arrival time), P(1), P(2), P(3) etc all at the same time - you are then double counting data. So, to answer your question, I don't think there is a more powerful test that can be constructed. – alpha137 Aug 6 '14 at 7:17