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Besides comparing the 1st moment, 2nd moment and 3rd moments, are there any other ways to demonstrate that two samples share the same distribution?

Perhaps use MLE and see if the parameters are similar? (Comparing parameters might become arbitrary though).

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    $\begingroup$ How would you show the samples to have the same first (population) moment? // Distributions can agree on all moments yet not be the same. $\endgroup$
    – Dave
    Commented Dec 6, 2022 at 19:26
  • $\begingroup$ Perhaps use MLE and see if the parameters are similar? Comparing parameters might be sort of arbitrary though. en.wikipedia.org/wiki/Maximum_likelihood_estimation $\endgroup$ Commented Dec 6, 2022 at 19:49
  • $\begingroup$ One of the best is a probability plot, because that shows you in detail how the sample distributions differ. Comparing moments is not a great method unless you are specifically interested in how the moments (rather than the distributions) differ. It's hard to see how comparing parameters would be "arbitrary:" that's exactly the same thing as comparing the distributions, under the basic MLE assumption that the distributions really do come from the parameterized family. $\endgroup$
    – whuber
    Commented Dec 6, 2022 at 21:34

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