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I'm validating a machine learning model that outputs a certain sample of N>20000. After fitting the sample to multiple distributions, and then running the typical Anderson-Darling and Kolmogorov-Smirnov tests, the only distribution that performs decently is the Johnson SU. I want to compute the confidence interval for the sample mean, assuming the underlying distribution is the Johnson SU one:

  • I know that I can compute confidence interval of certain statistics using bootstrapping too, please don't suggest it
  • I'm a student of engineering, with only 3 months of lessons in statistics- most of it i'm sure forgotten by now, so please try to explain it in a way that i can reproduce myself.

*Another question. Can I somehow use the transformation that links the standard normal distribution to this one? from the wikipedia Like, i get the x=0.05CDF(N[0,1]) and x=0.95CDF(N[0,1]), transform to the JohnsonSU through the formula and voila (mean = samplemean [+B,-C]@90% confidence)? Feels like im speaking nonsense, damn statistics...

Cheers and thanks in advance!

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  • $\begingroup$ if you do that transform, then your data owuld have mean 0 and sd 1 no? so that won't really help $\endgroup$
    – rep_ho
    Commented Nov 8, 2021 at 15:45

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If your sample is large, then you can use the usual formula for the CI of the mean because the sampling distribution will be normally distributed due to CLT. It's the sampling distribution that needs to be normally distributed, not the data, although that assumption is not extremely important anyway. If you are asking how to compute CI of the mean assuming a specific distribution of the data, then I do not know the answer, but you can always fit that distribution using Bayesian methods and get a credible interval from a posterior. I don't think it would make a difference in your situation.

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