Let X1, . . . , Xn ∼ iid Unif(0, θ). (a) Is this family MLR in Y= X(n)? (b) Find the UMP size-α test for H0 : θ ≤ θ0 vs H1 : θ >θ0. (c) Find the UMP size-α test for H0 : θ ≥ θ0 vs H1 : θ < θ0.(d) Letting R1 be the rejection region for the test in part (b) andR2 be the rejection region for the test in part (c). Consider thetest for the hypotheses H0 : θ = θ0 vs H1 : θ 6= θ0 determined bythe rejection region R = R1 ∪ R2. That is, we reject H0 if the datais in either R1 or R2. Find the power function of this test andcomment on the size.
