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My understanding is that for non-normal data, a widely accepted heuristic for using the t-test is a sample size of at least $n=30$. Since for smaller sample sizes the distribution of the sample mean will not be well approximated by a Student's t-distribution.

I was wondering what the corresponding heuristic is for using Hotelling's T$^2$ test.

To my mind the right metric to look at is the sample to feature ratio $\frac{n}{p}$. Since in the univariate case (t-test) this simply reduces to the sample size.

The reason I ask is because I have a sample size of approximately $n=20000$, however I have a feature size of approximately $p=5000$, which would put my sample to feature ratio at $\frac{n}{p}=4$, which doesn't seem like enough.

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    $\begingroup$ It depends on how non normal the data are for central limit theorem to apply. Mild non normality and 15 might be enough. Severe enough and thousands will be required. $\endgroup$ – Heteroskedastic Jim Aug 2 '18 at 3:40

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