I have a feeling this is a somewhat common problem but reading about tests for normal distributions hasn't really helped me as their assumptions didn't seem to fit mine.
Consider the following case:
- I have data (a lot) that is somewhat normally distributed but quite leptokurtic.
- I collect new data samples and want to test if the new data is from the same distribution.
What test should I use here? To give some context the testing would be done online (e.g. repeated after 10, 20 etc. samples) and I am interested to find out how many samples I need to reject the null hypothesis (where appropriate).
Edit: Alternatively, is there some way to forgo the distribution/goodness-of-fit testing and evaluate new incoming data according to its likelihood? I.e. data point 1 is in the 90-percentile of the distribution, point 2 in the 95% and so on. Eventually, I want a test to tell me that the behaviour is likely according to the previously collected data or not.