LMP Cauchy test: $\sum Y_i$ has null variance $n/8$, so the cut-off is $\sqrt{n/8}\,z_\alpha$

Let $X_1,\dots,X_n$ be iid Cauchy $C(\theta)$. The locally most powerful test of size $\alpha$ for testing $H_0:\theta=0$ against $H_1:\theta>0$ is (with $Y_i=X_i/(1+X_i^2)$):