Let $X_1,X_2,...,X_n$ be a random sample from $N(\mu,\sigma^2)$. Derive Uniformly Most Powerful test for $H_0: \sigma^2 \leq1$ vs $H_a: \sigma^2 \geq 2$.
I have been studying for qualifying exam and came across this problem. Since normal distribution has monotone likelihood property it can be easily shown that $H_0: \sigma^2 \leq1$ vs $H_a: \sigma^2 > 1$ is UMP by Karlin Rubin theorem.
But I could not derive UMP for $H_0: \sigma^2 \leq1$ vs $H_a: \sigma^2 \geq 2$.
Any help will be highly appreciated. Thanks.