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I would be thankful for some advise on this issue:
I want to test whether the skewness for two distributions is significantly different from each other. (I know that skewness is often not the best parameter to compare distributions but it is necessary in this case to go with skewness. The choice of another parameter is not possible.)
I have a large dataset on two different groups of individuals. Group 1 contains about 1 million observation for a certain continuous variable. Group 2 only about 6000. Also, the variance of the observations of group 1 is only about 1/3 of that of group 2. I would like to test whether the difference in skewness of the two population distributions is significant. However, I face the problem of huge sample size differences and also the differences in variances. Initially, I though about some non-parametric permutation test but I wonder whether this is a good choice in my setting. I guess not...
I would appreciate any comments on how to appropriately design a test.
Edit: This post does not solve my problem. The author assumes a certain distribution and does not face an issue with different sample sizes.