Distribution of the sample variance of n iid exponential variables

Question

I have to check some properties of an estimator, but I can't find its distribution.

Let $X_1,...,X_n $ be independent identically distributed exponential variables with parameter $ \theta $, i.e. with pdf $ f(x) = \frac{1}{\theta} \exp(-\frac{x}{\theta})$. What is the distribution of the sample variance $ \overline{X^2} - (\overline X)^2$?

What I found so far:

$\frac{2n}{\theta}\overline X$ ~ $ \Gamma(n,2)$

But how can I find the distribution of the sample variance?

Any help is greatly appreciated!

Answer 1

Your best bet is to use Monte Carlo simulation: repeatedly simulate $n$ draws from a exponential $\theta=k$ distribution, for some $k>0$, calculate the sample variance, repeat. Do this 10k-30k times to get the sampling distribution for the sample variance.

If you have data, then you can choose k to be the MLE for the exponential rate parameter.