$X_1,X_2$ are iid uniform ($theta, theta+1$). We consider the rejection region given by

{(x1, x2): x1 + x2 > C}$cup$

(

{(x1, x2): x1 > 1}$cup$

(

{(x1, x2): x2 > 1}.

What is the power function corresponding to this rejection region? This is from George Casella question 8.13 part (d). The solution is attached below. I don’t know how to analyze this new test is more powerful than the original rejection region $x_1 + x_2 > C$. If deriving the exact power function of the new rejection region is too hard, just comparison is also helpful. I don’t know how to compare.