MSE of the MME of $\theta$

Question

A random sample of size $n$ is taken from a gamma distribution with parameters $α = 8 $and $λ = 1/θ$. The sample mean is $\bar{x}$ and $θ$ is to be estimated.

Determine the mean square error of the MME of $θ$.

My attempt,

$$E(X)=\frac{8}{\frac{1}{\theta}}$$ $$\hat{\theta}{=\frac{\bar{x}}{8}}$$

$$Bias(\theta)=E(\hat{\theta})-\theta=0$$

$$MSE=Var(\frac{\bar{x}}{8})$$

Can I know how to proceed for it? Thanks in advance

Answer 1

The sum of i.i.d. Gammas is a Gamma. So in particular, $$ n\bar x = \sum_{i=1}^n X_i \sim \Gamma(8n,1/\theta).$$

So $$ \operatorname{Var}(\frac{\bar x}{8}) = \frac{1}{64 n^2}\operatorname{Var}(n\bar x) = ?$$