I have a parameterized, effectively black box process that generates a series of events (simulated action potentials). Different parameter values often lead to different numbers of events. How can I compare the Poissonness of the event timings across my parameter space? Is there a standard approach?
I am not looking for a single "best" parameter; I want to make a figure that shows how the Poissonness depends on the parameter choice. By "Poissonness" I mean the degree to which the output resembles that of a Poisson process, i.e. the degree to which the inter-event times are distributed according to an exponential distribution.
My approaches and their weaknesses
Originally, I just looked at the coefficient of variation of the inter-event interval. For a perfect Poisson process, this is near 1, but being near 1 does not mean we have a Poisson process.
I then thought I would try fitting an exponential curve to the top midpoints on a bar graph, but that seems unjustified and in any case loses information by aggregating data.
To avoid losing information, I could use the maximum likelihood estimate and quantify the Poissonness according to the likelihood. The problem here is that the likelihood is linked to the number of events, so I cannot directly compare likelihoods from different event trains. I can take the $n$th root of the likelihood (where $n$ is the number of events); intuitively this feels fair, but I'm not sure if it is statistically justified. Is there a better way?