Let $X\_1,...,X\_n$ be a random sample from $N(0,\sigma^2)$, where $\sigma>0$ is unknown. We try to estimate $\sigma$ using $T\_1=\sqrt{\frac{\pi}{2}}\frac{1}{n}\sum^n\_{i=1}|X\_i|$ and $T\_2=\sqrt{\frac{1}{n}\sum^n\_{i=1}(X\_i^2)}$. What is the asymptotic relative efficiency of $T\_1$ w.r.t $T\_2$?

**my work:**

If these two estimators are such that $T\_1 \sim AN(\tau(\theta),\sigma^2\_1/n)$ and $T\_2 \sim AN(\tau(\theta),\sigma^2\_2/n)$, then $ARE(T\_1,T\_2)=\sigma^2\_2/\sigma^2\_1$.

I am having troubles finding the asymptotic distributions of $T\_1,T\_2$. I believe that I will need to use the Delta Method and properties of the half-normal distribution, since $T\_1$ makes use of the random variable $|X|\sim HN(\sigma)$.

I have made some slight progress in finding the asymptotic distribution of $T\_2$. By CLT, $\frac{1}{n}\sum(X\_i^2)\sim AN(\sigma^2,Var(X^2)/n)$. Then, by letting $g(z)=\sqrt{z}$, Delta Method yields $g(\frac{1}{n}\sum(X\_i^2))=T\_2 \sim AN(\sigma, \frac{Var(X^2)}{n}\cdot\frac{1}{4\sigma^2})$. However, I do not know how to evaluate $Var(X^2)$.

I used a similar approach for finding the asymptotic distribution of $T\_1$. Since $|X| \sim HN(\sigma)$, we have $\bar{|X|} \sim AN(\frac{\sigma\sqrt{2}}{\sqrt{\pi}},\frac{\sigma^2(1-2/\pi)}{n})$ from CLT. Then, we use Delta Method with $g(z)=z\sqrt{\pi/2}$ to get $T\_1 \sim AN(\sigma, \frac{\sigma^2\pi(1-2/\pi)}{2n})$. How can I complete this problem?