Non-parametric test for difference in parameter with unequal sample size and different variance [duplicate]

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.

Thanks, Frank

Edit: This post does not solve my problem. The author assumes a certain distribution and does not face an issue with different sample sizes.

marked as duplicate by Alexis, kjetil b halvorsen, Peter Flom♦Mar 24 '18 at 13:50

• @Alexis: Thanks for your comment. No, it's not a duplicate post. I've read this post before and it does not help in my case. It is also not from me. This is about 2.5 years old. – Frank Mar 23 '18 at 18:32
• Please give us your quantitative definition of "skewness." There are several common ones but they differ. – whuber Mar 23 '18 at 18:49
• I use the "textbook" skewness: 3rd central moment scaled by the standard deviation to the power of 3. – Frank Mar 23 '18 at 18:57

My first thought would be to do a monte carlo simulation to bootstrap an empirical P-value.  (My mind went here first as you asked about a non-parametric possibility...and I don't recall ever working with a nonparametric parallel statistic for skewness.)

First, you calculate the difference in the skewness between your two samples.

Second, you shuffle the data together, split the shuffled data into two comparably sized data sets, and recalculate the difference in the skewness.

Third, repeat second step 10,000 times.  (Sounds tedious, but a program like R can handle this without too much pain or sorrow.)

Fourth, determine where your observed statistic from the first step falls in the distribution of simulated statistics.

Happy to elaborate further (or provide assistance with coding something like this if it seems like an approach you'd like to pursue).

• Thanks a lot Gregg. :) I guess, I will follow your advise. I will go with Matlab, which I am quite familiar with. – Frank Mar 24 '18 at 0:45