Let Xi iid N(, 2 ), i = 1, . . . , n. Let 2 = P(Xi) 2 n . We showed early in the semester that E ( 2 ) = 2 and V ( 2 ) = 2 4.

(a) (5 points) Can there be an unbiased estimator of 2 that has a smaller variance than 2 ? Prove using the Cramer-Rao inequality. Hint: It might be easier to let = 2 , and = 2 .

(b) (2 points) What is MSE ( 2 )

(c) (5 points) Consider the biased estimator 2 2 = P(Xi) 2 n+c for some constant c. Find the MSE ( 2 2 ) (this will be a function of c). Find the value of c that minimizes MSE ( 2 2 ). For this c, what is MSE ( 2 2 ).

(d) (1 point) Which is the "better" estimator of 2 : 2 or 2 2 ? Defend your answer.

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