Suppose X₁, ... , X_n are i.i.d. with density function

f(x) = (1 + )xy (0 x 1)

The densities are on the unit interval and the unknown parameter is non-negative.

(i) For testing the null hypothesis H₀ : = 1 against the simple alternative hypothesis H₀ : = 1, determine the most powerful level test as explicitly as possible. (Here, ₁ is fixed and specified.)

(ii) Describe how you would compute a critical value for the test in (i). Be as explicit as possible, even if your solution is an approximation.

(iii) Does there exist a UMP level alpha test for testing H₀ : = 1 against the two-sided alternative H₁ : 1?

(iv) For the two-sided problem in (iv), find a generalized likelihood ratio test as explicitly as possible.