Let $X_1, ..., X_n$ be a random sample from the Pareto distribution with pdf
$$f(x|\theta) = \begin{cases} \theta x^{-(\theta+1)}, & x\geq 1 \\\ 0, & 1 < x \end{cases} $$
where $\theta > 0$ is unknown.
Find a uniformly most powerful (UMP) test of size $\alpha$ for testing
$H_0: \theta \leq \theta_0$ versus $H_1: \theta > \theta_0$ where $\theta_0 > 0$ is a fixed real number.
Using Neyman-Pearson, the mp test is $\phi(x) = \begin{cases} 1, & \frac{f(\bf x, \theta_1)}{f(\bf x, \theta_0)} > k \\\ 0, & else \end{cases} $
By solving for $\alpha = P_{H_0}(\sum_{ i = 1}^{n} \ln(X_i)) => 1- \alpha = P(2\theta\sum_{ i = 1}^{n} \ln(X_i) > 2\theta_0k) $
I get $k = \frac{1}{2\theta_0}\chi^2_{2n,1-\alpha}$
and $\phi(x) = \begin{cases} 1, & \sum_{ i = 1}^{n} \ln(X_i) < \frac{1}{2\theta_0}\chi^2_{2n,1-\alpha} \\\ 0, & else \end{cases} $
as the mp test.
I'm unsure how to go about showing whether this is the UMP test (rather than only MP). Any ideas? Also I'd really appreciate a comment on the correctness of the approach. I'm new to this sort of mathematics/statistics.