Given $X_1,...,X_n \sim \text{Exp}(\theta)$ and $Y_1,...,Y_n \sim \text{Exp}(\frac{1}{\theta})$, where $\theta>0$ and have the same $\theta$ in both distributions. $X$ and $Y$ are independent.
I have found the MLE of the simultaneous distribution: $$\hat{\theta}=\sqrt{\frac{X_{\bullet}}{Y_{\bullet}}},$$ where $X_\bullet=\sum_{i=1}^n{x_i}$ and $Y_\bullet=\sum_{i=1}^n{y_i}$.
I know that I can find the exact distribution, which is: $$\hat{\theta}\theta^{-1}\sim \sqrt{F},\quad\quad F\sim F_{2n,2n},$$ however, my professor insist that I has to find the asymptotic distribution of the MLE. Please see the question here, where the focus were on why the asymptotic solution not were the best, however, I still has to find it.
I thought that I could use CLT in combination with the Delta Method, but I really don't see a proper fit.
Any help will be greatly appreciated.
EDIT
This question was part of an old exam set at my university, and the professor who made the exam set (whom don't work as a professor today), have provided me with the official solution, which he still had:
As in the linked question, here, I had found the correct asymptotic distribution. It is:
$$\hat{\theta} \sim \mathcal{N}\left(\theta,(ni_\theta)^{-1}\right),$$ where $i_\theta$ is the Fisherinformation.