Confidence intervals for the variance. What if data is not normally distributed? I am writing an essay about confidence intervals for the variance and there is a lot of information available under the assumption that our data is normally distributed, but there is not much said about other possible distributions.
How can we find confidence intervals for the variance if we assume our data is e.g. uniformly distributed? Can it be done? Can it be done for other distributions?
 A: In cases where the distribution of the sample variance can be determined (such as for the normal distribution, where it follows a scaled Chi-squared distribution). But it turns out this is pretty hard, even for the uniform distribution. 
The paper Exact distribution of the sample variance from a gamma parent distribution by Thomas Royen manages to do it for a Gamma distribution, however. Also, Hyrenius did it for a mixture of two normal distributions (in Distribution of Student–Fisher’s t in Samples from Compound Normal Functions).
On the other hand, one can always use the CLT to show the the distribution of the sample variance converges to a Gaussian distribution under certain conditions (see here, here, and here), which can then be used to construct the confidence intervals. 
Lastly, it's worth noting that the first two moments of the sample variance (for iid variables), even for non-normal distributions, can be written:
$$
\mathbb{E}[s^2]=\sigma^2\;\;\;\&\;\;\;\mathbb{V}[s^2]=\frac{1}{n}
\left( \mu_4 - \frac{n-3}{n-1}\sigma^4 \right)
$$
where $\mu_4$ is the fourth centralized moment.
