Let $X_1,...,X_n$ be independent and identically distributed from a $N(\theta,\theta)$ distribution (implying that the variance of $X_i$ is $\theta$), where $\theta>0$ and $\theta$ is an unknown parameter.
Suppose that we want to estimate the quantity $\theta(\theta-1)$.
Determine the asymptotic relative efficiency of $\bar X(\bar X-1)$ with respect to $\delta_n=\sum x_i^2/n$.
So essentially this means that I have to compute the quantity:
$$\lim_{n\to\infty}\frac{V(\bar X(\bar X-1))}{V(\delta_n)}$$
However, this is where I'm stuck; I'm not sure how to evaluate either of these variances, but especially not the numerator. After all, $\bar X$ is definitely NOT independent of $(\bar X)^2$. I suspect that I may need to use the Stein identity since the random variables are normally distributed; but I'm honestly not sure how to apply it to this situation.
How can I proceed from here?
$Y=\bar X$ is normally $N(\theta, \frac{\theta}{n})$ distributed. You can simply find $$\text{Var}(Y(Y-1))=\mathbb E[Y^4-2Y^3+Y^2]-\left(\mathbb E[Y^2]-\mathbb E[Y]\right)^2$$ and substitute moments of $Y$ from the table https://en.wikipedia.org/wiki/Normal_distribution#Moments .