In his answer to my previous question, @Erik P. gives the expression
$$
\mathrm{Var}[s^2]=\sigma^4 \left(\frac{2}{n-1} + \frac{\kappa}{n}\right) \>,
$$
where $\kappa$ is the excess kurtosis of the distribution. A reference to the Wikipedia entry on the distribution of the sample variance is given, but the wikipedia page says "citation needed".
My primary question is, is there a reference for this formula? Is it 'trivial' to derive, and if so, can it be found in a textbook? (@Erik P. couldn't find it in Mathematical statistics and data analysis nor I in Statistical Inference by Casella and Berger. Even though the topic is covered.
It would be nice to have a textbook reference, but even more useful to have a (the) primary reference.
(A related question is: What is the distribution of the variance of a sample from an unknown distribution?)
Update: @cardinal pointed out another equation on math.SE:
$$
\mathrm{Var}(S^2)={\mu_4\over n}-{\sigma^4\,(n-3)\over n\,(n-1)}
$$
where $\mu_4$ is the fourth central moment.
Is there some way that to rearranged the equations and resolve the two, or is the equation in the title wrong?
Best Answer
Source: Introduction to the Theory of Statistics, Mood, Graybill, Boes, 3rd Edition, 1974, p. 229.
Derivation: Note that in the OP's Wikipedia link, $\kappa$ is not the kurtosis but the excess kurtosis, which is the "regular" kurtosis - 3. To get back to the "regular" kurtosis we have to add 3 in the appropriate place in the Wikipedia formula.
We have, from MGB:
$\text{Var}[S^2] = {1\over{n}}(\mu_4 - {{n-3}\over{n-1}}\sigma^4)$
which, using the identity $\mu_4 = (\kappa + 3)\sigma^4$, can be arranged to (derivation mine, so any errors are too):
$ = {1\over{n}}(\kappa \sigma^4 + {{n-1}\over{n-1}}3\sigma^4 -{{n-3}\over{n-1}}\sigma^4) = \sigma^4\left({\kappa \over{n}}+{3(n-1)-(n-3)\over{n(n-1)}}\right) = \sigma^4\left({\kappa\over{n}} + {{2}\over{n-1}}\right) $