Question: Do someone could give a solution to this question? 4. Let X = X1, X2, ..., Xn be i.i.d. random variables, each with a density

Do someone could give a solution to this question?

4. Let X = X1, X2, ..., Xn be i.i.d. random variables, each with a density f (a; 0) = 2 0 , x > 0,0>0 0 elsewhere a) Prove that the the family has a monotone likelihood ratio in the statistic T = ELIX?. b) Show that Y = 2X? has a x? distribution with a density fy(y) = zexp ( -!), y > 0. c) Using a) and b), derive completely the uniformly most powerful size-a test * of Ho : 0 60 (Go is a specified constant). Sketch a graph of its power function. d) For testing Ho : 0 = 60 = 2 versus H1 : 0 # 2, a uniformly most powerful unbiased test exists. Formulate the structure of the test (do not determine the rejection constants)

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