Date: Apr 8, 2014 6:53 PM
Author: Luis A. Afonso
Subject: Where the lines go cross each other . . .
. . we find the minimum size two samples should have in order they are distinguishable through a common parameter.

When we successively set sample sizes to increase the skewness of Gumbel and Normal Distributions behave as such: do decrease the 95% quantile of the former, increases the 5% of the latter. Obtained from 40ยด000 random sample each size.

The following results are quite eloquent:

Skew 95% fractile__________Skew 5% fractile

Normal/Gaussian___________Gumbel (A=0, B=1)

n=74____0.448_______________0.378____

n=76____0.441_______________0.393____

n=78____0.434_______________0.391____

n=80____0.431_______________0.397____

n=82____0.427_______________0.405____

n=84____0.422_______________0.416____*

n=86____0.417_______________0.425____*

n=88____0.412_______________0.427____

n=90____0.408_______________0.434____

Conclusion: in order that a Gumbel sample be distinct by its skewness from Gaussian ones one must chose sizes sample sizes somewhere at 84 - 86.

Repeating (as expected to perform in simulative procedures):

Skew 95% fractile__________Skew 5% fractile

Normal/Gaussian___________Gumbel (A=0, B=1)

n=74____0.449_______________0.379____

n=76____0.439_______________0.383____

n=78____0.437_______________0.396____

n=80____0.433_______________0.399____

n=82____0.425_______________0.411____

n=84____0.422_______________0.415____*

n=86____0.416_______________0.422____*

n=88____0.412_______________0.432____

n=90____0.412_______________0.436____

Luis A. Afonso