Given distribution function: $f(x) = \theta x^{\theta - 1}, \text{ if } 0<x<1 \text{ and } 0 \text{ otherwise}$. 1) Test hypothesis $H_0 = \theta = \theta_0 $ with generalised likelihood ratio test.
2) Prove than null hypothesis $H_0$ is rejected, when $$-2n\left[ln\left( \frac{\theta_0}{\hat{\theta}}\right)-\left(\frac{\theta_0}{\hat{\theta}}-1\right) \right] \ge \chi_{1-\alpha, 1}^2, \text{ where } \hat{\theta} \text{ is maximum likelihood estimator for } \theta$$
Likelihood function: $\frac{n}{\theta}+ \Sigma^n_{i=1} ln(x_i)\hat{\theta} = nln(\theta) + \Sigma^{n}_{i = 1} (\theta - 1)ln(x_i)$
What is the procedure to prove this? ( for hypothesis testing I have to use generalized likelihood ratio test: $-2ln \lambda(x)$)