I wish to prove for $X_1,\dots,X_n$ i.i.d. Cauchy($\theta,1$) random variables that there doesn't exist a UMP for $H_0: \theta = 0$ against $H_1: \theta>0$.
For $\theta_1<\theta_2$, NP tests $(\varphi_1)$ for $H_0: \theta = 0$ against $H_1: \theta=\theta_1$ and $(\varphi_2)$ $H_0: \theta = 0$ against $H_1: \theta=\theta_2$ should produce the respective critical regions $$\prod_{i=1}^n \frac{1+(x_i-\theta_1)^2}{1+x_i^2} < k$$
and
$$\prod_{i=1}^n \frac{1+(x_i-\theta_2)^2}{1+x_i^2} < k$$
I'm at a bit of a loss how to proceed from here. How do I show with rigor that these regions are not equal?