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Suppose $X$ is distributed as standard normal, I take a sample of size $n$, and compute 3rd and 4th sample cumulants $\kappa_3$ and $\kappa_4$. I'm interested in the asymptotic distribution of $\kappa$'s as $n\to \infty$

I'm assuming $\sqrt{n}\kappa_3$ and $\sqrt{n}\kappa_4$will approach normal distribution, what is the mean and variance?

Yaroslav Bulatov
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    I don't have time tonight to give the details but the expectations of the sample cumulants (3 and 4) are 0 and $-\frac{6 (n-1)}{n^2}$, respectively. The variances are $\frac{6 (n-2) (n-1)}{n^3}$ and $\frac{24 (n-1) ((n-6) n+24)}{n^4}$, respectively. – JimB Sep 05 '20 at 06:02
  • Thanks! Is there a standard reference that derives this? – Yaroslav Bulatov Sep 05 '20 at 15:48
  • Probably but I used *Mathematica* much as you did in https://mathematica.stackexchange.com/questions/229568/expressions-for-moments-of-sample-cumulants and did some guessing with help from the `FindSequenceFunction`. – JimB Sep 05 '20 at 18:34
  • I can't get hold of a copy at the moment to check, but I believe that Vol I of Kendall and Stuart (*Advanced theory of Statistics*) / or more recently Stuart and Ord (*Kendall's Advanced Theory of Statistics*) has what you seek. – Glen_b Sep 06 '20 at 10:36
  • Thanks. I found scanned version of 1952 edition, but the section on cumulants didn't have it, so it must be one of the newer editions – Yaroslav Bulatov Sep 06 '20 at 16:01
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    Found a Mathematica package that can calculate this, the "Moments of moments" example here http://www.mathstatica.com/examples/MOM/index.html – Yaroslav Bulatov Sep 07 '20 at 19:14
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    *mathStatica* is a great package. Note that *mathStatica* uses the unbiased estimators of the cumulants, whereas the *Mathematica* function `Cumulant` does not provide an unbiased estimator for every cumulant. – JimB Sep 07 '20 at 21:36

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