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**Unformatted text preview: **PROF HONG FALL 2011 ECONOMICS 3190 PROBLEM SET #11 1. Suppose ^ & 1 ; ^ & 2 and ^ & 3 are estimators of &; and we know that E ( ^ & 1 ) = E ( ^ & 2 ) = &; E ( ^ & 3 ) 6 = &; var ( ^ & 1 ) = 12 ; var ( ^ & 2 ) = 10 ; and E ( ^ & 3 & & ) 2 = 6 : Which estimator is the best in terms of MSE criterion? 2. Let X 1 ; X 2 ; X 3 ; X 4 be an i.i.d. random sample from a population with ( ; 2 ) : Consider the following two estimators for : ^ 1 = X 1 + 2 X 2 + 3 X 3 + 4 X 4 10 ; ^ 2 = X 1 + 4 X 2 + 4 X 3 + X 4 10 : Which estimator, ^ 1 or ^ 2 ; is more e cient? 3. A random sample, X 1 ; :::; X n ; is taken from an i.i.d. population with ( ; 2 ) : Consider the following estimator of : ^ = 2 n ( n + 1) n X i =1 i X i = 2 n ( n + 1) ( X 1 + 2 X 2 + 3 X 3 + + nX n ) : (a) Show ^ is unbiased for : (b) Which estimator, ^ or & X n ; is more e cient? [ Hint: P n i =1 i = n ( n +1) 2 and P n i =1 i 2 = n ( n +1)(2 n +1) 6 : ] 4. Suppose f X 1 ; X 2 ; :::; X...

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