Question: Question Four Let X = (X1, X2, ..., Xn) be i.i.d. random variables, each with a density f (x, 0) = - A exp 20

Question Four Let X = (X1, X2, ..., Xn) be i.i.d. random variables, each with a density f (x, 0) = - A

Show that the family L(X, 0) has a monotone likelihood ratio in

Question Four Let X = (X1, X2, ..., Xn) be i.i.d. random variables, each with a density f (x, 0) = - A exp 20 where 0 > 0 is a parameter. a) [2 marks] Show that the family L(X, 0) has a monotone likelihood ratio in T = Et, X?. b) [2 marks] Argue that there is a uniformly most powerful (UMP) or-size test of the hypothesis Ho : 0 60 and write down its structure. You do not need to find the threshold constant that defines the test in this part. c) [2 marks] Using the density transformation formula (or otherwise) show that Yi = has density function: fr(y) = -e y/2 y>0. That is, Yi ~ X2.d) [2 marks] Using c) (or otherwise), find the threshold constant in the test and hence determine completely the uniformly most powerful a- size test 4* of Ho : 0 60- e) [2 marks] Calculate the power function End* and sketch a graph of the power function as precisely as possible

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