I used the inter-arrival time data to fit a Poisson process. But some one asked to check the independence for the data. For example, if I have a inter-arrival time set {2,3,4,5,4,9}.
How to test independence?
I don't think that there is a way to test independence (except the independence of frequency distributions with the Chi² independence test). Independence of the observations is an assumption made under which the likelihood can be calculated. This should be assured by the way the data is obtained.
In time-series analyses one may have a look at auto-correlation structures to see if there is some dependency over time.
To Jochen Wilhelm, thanks. Perhaps using auto-correlation is the only way to tell the independencies some how.
To Fausto Galetoo, Thanks for letting me know about the properties of poisson and regenerative processes. But still how do we know about the independencies between different inter arrival times?
Before doing a fit of data to a certain probability distribution, it is recommended to check if the data are independent identically distributed (i.i.d.). Therefore I would first check stationarity to cover the "identical" property using a unit root test (e.g. Dickey-Fuller) and for "independence" the BDS-test (Brock Dechert Scheinkman 1987).
Because the original BDS-paper had huge delays in its publication, every practitioner uses a preprint version of it (see enclosed pdf). The maths of the BDS-paper is essentially an auto-correlation analysis with a Gaussian confidence test of the correlations being either significantly different from 0 or not.
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