Let $X_1, . . . , X_n$ be a random sample from $X$ with density function $f(x; \theta) = 2 \theta^{-1}xe^{-x^2/\theta} \cdot 1_{\{x>0\}}$ where $\theta > 0$. By using an appropriate approximation, find the generalized likelihood ratio test of size $\alpha$ for the testing problem $H_0 : \theta=\theta_0$ vs. $H_1 : \theta\neq\theta_0$.
I'm very new to testing in statistics, so I have no idea how to even begin this problem.
$\textbf{EDIT: }$ I've found the MLE to be $$\hat{\theta} = \frac{1}{n}\sum^n_{i=1}x_i^2.$$ Now computing the generalized likelihood ratio function we get $$\lambda_n = \left(\frac{\hat{\theta}}{\theta_0}\right)^n \exp\left(-n\left(\frac{\hat{\theta}}{\theta_0}-1\right)\right).$$ This means that $$-2n\ln\left(\frac{\hat{\theta}}{\theta_0}\right) + 2n \left(\frac{\hat{\theta}}{\theta_0}\right)$$ goes to a $\chi^2(1)$ in distribution. How do I proceed from here?