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homework3_2011

homework3_2011 - PROF HONG FALL 2011 ECONOMICS 3190 PROBLEM...

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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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homework3_2011 - PROF HONG FALL 2011 ECONOMICS 3190 PROBLEM...

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