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I think the most popular alternative to the "normal" Kolmogorow-Smirnow-Test for normal distribution (if the parameters are unknown and therefore have to be estimated) is the Lilliefors-Test. I am thinking about this (maybe too simple) alternative: first, you calculate the empirical mean and empirical variance of the given data (or of a subset of it). Afterwards, you generate a sample of normally distributed random variables with these estimated parameters. Then, you can test, whether the original sample and the generated sample have the same distribution using the 2-sample-kolmogorow-smirnow-test.

This would be a simple method to check, whether a given sample follows a normal distribution. I don't see problems but it seems to be too simple. Does anybody see problems/mistakes in this idea?

Thank you very much!

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