Suppose I have a sequence of iid random variables $X_1, \ldots, X_n$ following the pdf:
$$ f_\theta (x) = \theta x^{\theta-1} $$
for $\theta >0$ and $0 <x<1$.
I would like to obtain a level-$\alpha$ likelihood ratio test for the null hypothesis $H_0: \theta = \theta_0$ versus the two-sided alternative $H_1: \theta \neq \theta_0$ where $\theta_0$ is a known constant.
MY ATTEMPT:
I first construct the ratio:
\begin{align} \lambda(x) &= \dfrac{sup_{\theta=\theta_0}L(\theta|X)}{sup_{\theta\neq\theta_0}L(\theta|X)} \\ &= \dfrac{\theta_0^n \left(e^{\sum log x_i}\right)^{\theta_0-1}}{\left(\frac{n}{-\sum logx_i}\right)^n \left(e^{\sum logx_i}\right)^{\left(\frac{n}{-\sum log x_i} - 1\right)}} \\ &= \left(\dfrac{-\theta_0 \sum log x_i}{n}\right)^n e^{n+\theta_0 \sum log x_i} \end{align}
The denominator is calculated using the MLE of $\theta$ which is $\theta^{MLE} = \frac{n}{-\sum log x_i}$. Now, I'd like to find the likelihood ratio test, in that I would like to choose a constant $c$ such that:
$$ \alpha = sup_{\theta = \theta_0}P_\theta\left(\lambda(x) \leq c\right) $$
Now, $-\sum logx_i \sim Gamma(n, \theta)$ but I CANNOT isolate the above equation due to the $log$ form. What is the right answer here? Thanks!