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Unformatted text preview: PROF HONG FALL 2006 ECONOMICS 619 FINAL EXAM Notes: (1) This is a closed book/notes exam. There are 7 questions, with a total of 100 points; (2) you have 150 minutes; (3) suggestion: have a look at all problems and ...rst solve the problems you feel easiest; (4) good luck! 1. [15 pts] Suppose Y = + X + jXj"; where E(X) = 0; var(X) = 2 ; E(") = 0; var(") = 2 ; and " and X are independent. " X Both and are constants. (a) [5 pts] Find E(Y jX): (b) [5 pts] Find var(Y jX): (c) [5 pts] Show cov(Y; X) = 0 if and only if = 0: 2. [10 pts] Suppose fXi gn is an i.i.d.N ( ; 2 ) random sample, where both and 2 i=1 are unknown parameters. Contruct an unbiased estimator for = 2 , and justify it is unbiased. 3. [15 pts] (a) Suppose X n = fXi gn is an i.i.d. random sample with a population i=1 probability density f (x; ): Is X n a su cient statistic for ? Give your reasoning. (b) Suppose X n = fXi gn is an i.i.d. random sample from a N ( ; ) population, i=1 where is unknown. Find a su cient statistic for . Give your reasoning. 4. [20 pts]: Suppose fXi gn are i.i.d.N (0; 2 ): There are two estimators for 2 : i=1 ^2 1 ^2 2 1X 2 = X ; n i=1 i
n n P where X = n 1 n Xi : i=1 (a) [5 pts] Check whether ^ 2 and ^ 2 are unbiased for 1 2 1 1X = (Xi n i=1 X)2 ; 2 . Give your reasoning. (b) [15 pts] Which estimator, ^ 2 or ^ 2 ; is more e cient in terms of mean squared 2 1 error? Give your reasoning. 5. [10 pts] Suppose a sequence of random variables fZn g is de...ned as Zn PZn 1
1 n 1 n n
1 n (a) [4 pts] Does Zn converges in mean square to 0? Give your reasoning clearly. (b) [6 pts] Does Zn converges in probability to 0? Give your reasoning clearly. 6. [10 pts] Let X n = fX1 ; X2 ; :::; Xn g be an independent but not identically distributed random sample with E(Xi ) = and V ar(Xi ) = 2 =i2 ; where i = 1; 2; :::; n: Both and 2 are unknown. De...ne a class of estimator for as ^=
n X i=1 ci Xi : N (ai ; 7. [20 pts] Suppose fXi gn is an independent random sample and Xi i=1 2 s, i = 1; :::; n; where ai and i are known constants that dier across dierent i' and an unknown parameter. Thus, the probability density of Xi is f (xi ; ) = p 1 2
2 i P (a) [4 pts] Show that ^ is unbiased if and only if n ci = 1: i=1 (b) [6 pts] Find the most e cient unbiased estimator ^ from the class of ^ : P [Hint: n i2 = n(n + 1)(n + 2)=6:] i=1 2 i ); is exp (xi 2 ai )2
2 i : (a) [10 pts] Find the MLE ^ for ; and check if it is a global maximizer. (b) [10 pts] Does ^ achieve the Cramer-Rao lower bound? Give your reasoning. 2 ...
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