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Unformatted text preview: Sample problems for midterm 1 There will be 3 problems on the test similar to the problems below. 1. Which of the following estimators for the mean is the most efficient, i.e. has the smallest MSE? Assume that = 2 . a) X 1 + X 2 2 . b) X 1 + X 2 + X 3 + X 4 5 c) X 1 X 2 + X 3 X 4 + X 5 d) X 1 + X 2 3 e) X 1 +2 X 2 + X 3 4 . Solution: Using the result, that for any estimator its MSE is equal to the sum of the square of the bias and the variance, we obtain that the MSE is, respectively: a) 0+ 2 2 = 2 2 ; b) 2 25 + 4 2 25 = 8 2 25 ; c) 0 + 5 2 = 5 2 ; d) 2 9 + 2 2 9 = 6 2 9 = 2 2 3 ; e) 0 + 6 2 16 = 3 2 8 . Thus we see that the estimator in b) has the lowest MSE. 2. Consider a continuous distribution with the pdf f ( x ) = 1 2 xe x/ . Find the MLE of the parameter based on a random sample of size n . Solution: The likelihood function has the form L ( ) = 1 2 n n Y i =1 x i !...
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This note was uploaded on 02/26/2008 for the course IE 121 taught by Professor Perevalov during the Spring '08 term at Lehigh University .
 Spring '08
 Perevalov

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