Assume that the hypothesis H0 : = 0 is to be tested against H1 : > 0, where is a parameter in

Assume that the hypothesis H0 : θ = 0 is to be tested against H1 : θ > 0, where θ is a parameter in the U[θ,θ + 1] distribution. (i) Derive the test of size α = 0.05, based on one observation only, and obtain its power function for θ = 0.1, 0.2, ..., 0.9.

(ii) Use Monte Carlo simulations to obtain a test with a critical region X3 > k; that is, estimate k. Then again use Monte Carlo simulations to estimate the power function of this test at θ = 0.1, 0.2, ..., 0.9. Use the same size α = 0.05. (iii) Compare the power functions of both tests.