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I understand that one of the basic assumptions of a Poisson process is that in a small enough interval, the probability of more than one arrival is negligible.

In my case, I have data that shows the number of arrivals per day. Most days contain no arrivals, some have 1, few have 2, but there is a single occasion where there are 10 arrivals in one day. Since I don't have the precise times at which these 10 arrivals occurred, this doesn't necessarily violate one of the Poisson assumptions that I stated above, does it?

How should I treat this situation? If I fit a Poisson distribution to the count data, then that anomaly day will cause $\lambda$ to be larger than it would otherwise. On the other hand, that 10-day is rare, but also somewhat explainable given where the data comes from. I.e. I think the arrivals follow a Poisson process, but occasionally they will come in a "cluster".

Can you please give me some advice on how these situations are usually dealt with? I feel there are really two questions lurking underneath here: one having to do with discretizing data, and the other to do with compound Poisson processes. But these two concepts are blurred together for me.

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  • $\begingroup$ It may be useful to rephrase the question or make it more precise in case that the provided answer does not help you. $\endgroup$ – mloning Aug 17 '18 at 14:00
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Discretising time: As you noted, you do not directly observe the time-stamped data from the point process, but rather only the counts of events for given time bins, in your case, days. So, the assumption that you state does not direclty relate to the the data that you observe and cannot be tested by it. So, fitting a Poisson distribution still seems appropriate.

Compound Poisson process: If you observe counts only, it's not a compound Poisson process. A compound Poisson process is formed by adding (independent and identically distributed) non-negative random values to each point of the original Poisson point process. The compound Poisson process is then formed from the sum of all the random values. For example, you may observe the money spent by each customer in addition to the arrivals.

It's hard to give more specific advice without knowing what you're trying to find out. Generally, point processes are useful when working with time-continuous rather than time-binned data. For time-binned data, time-series analysis is a good starting point.

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