Question: (d) Let X = {X1, X2,..., X,} be a random sample of size n from the following probability density function: 1 fx(x; 6) = ;

(d) Let X = {X1, X2,..., X,} be a random sample

(d) Let X = {X1, X2,..., X,} be a random sample of size n from the following probability density function: 1 fx(x; 6) = ; ) = 2e-/0 for 2 > 0, and 0 otherwise. i. Derive the maximum likelihood estimator of 0. (You do not need to verify the solution is a maximum.) (7 marks) ii. Show that the estimator derived in part i. is consistent for 6. Hint: You may use the fact that E(X) = 20 and Var(X) = 202. (6 marks) iii. Derive the total Fisher information and hence state the asymptotic sampling distribution of the estimator derived in part i. (7 marks)

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