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Suppose I have a sample of n variables and I know that the population mean is 0 and variance is 1. I want to check if the sample skew defined as

$$ s=\frac{1}{n} \sum_i X_i^3 $$

is statistically significantly different from that the population distribution being unit normal. My way of doing it was to say under a standard normal assumption

$$ E[s]=0 \\ E[s^2]= \frac{15}{n} $$

so I check the p-value of my sample s against a $\mathcal N(0,\sqrt{15/n})$. However scipy.stats has a function to test the significance of skewness against a standard normal and it looks quite different from what I have (see source following the link https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.skewtest.html#scipy.stats.skewtest). They do not provide a reference. Now obviously the distribution of $X^3$ is not Gaussian but for large enough $n$ CLT should make the tests the same but I was wondering if anyone knows what test scipy.stats is using or any other test for skewness.

Borun Chowdhury
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  • I don't know or remember a reference, but you can check the references in scipy's normal_test, D'Agostini articles, to see whether it came from them or is older than that. – Josef Jan 10 '17 at 04:26
  • The only test of skewnes I could find so far ist based on: D'Agostino, R.B. (1970). Transformation to Normality of the Null Distribution of G1. Biometrika, 57, 3, 679-681. – stollenm Jun 20 '18 at 11:57
  • Some related posts: https://stats.stackexchange.com/questions/182472/testing-for-significant-differences-in-skewness, https://stats.stackexchange.com/questions/425504/hypothesis-testing-skewness-in-two-sample-cases – kjetil b halvorsen Jun 22 '22 at 23:15

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