I was reading the section on k-statistics on wolfram alpha. It was known to me that for the sample variance
$k_2 = \frac{1}{n-1}\sum_{i=1}^n (x_i - \overline{x})^2$
it holds that its variance equals
$var(k_2) = \frac{\kappa_4}{n} + \frac{2 \kappa_2}{n-1} = \frac{\mu_4}{n} - \frac{\sigma^4(n-3)}{n(n-1)},$
where $\kappa_i$ denotes the $i$-th cumulant, $\mu_4$ the 4-th central moment and $\sigma^2$ the variance.
Now, apparently there exists an unbiased estimator for $var(k_2)$ given by
$\hat{var}(k_2) = \frac{2n k_2^2 + (n-1)k_4}{n(n+1)},$
where
$k_4 = \frac{n^2}{(n-1)(n-2)(n-3)}\left( (n+1) m_4 - 3(n-1) m_2^2 \right)$.
Here $m_p = \frac{1}{n} \sum_{i=1}^n (x_i - \overline{x})^p$.
There reference given is Kenney and Keeping 1951, p. 189. However, I cannot find a copy anywhere, or a derivation of this equation.
Can anyone help me with this derivation or point me towards a reference?
Also, I was wondering if a similar equation would provide an unbiased estimator for the variance of the sample covariance.
