L04_gaussmarkovans

L04_gaussmarkovans - (f) The estimator is not linear...

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Answers to exercises in: The Gauss-Markov Theorem Stat 640 Answer 1 (a) The estimator is linear with a = (1 / 2 , - 1 / 2 , 0 ,..., 0) 0 and unbiased because E [( y 1 + y 2 ) / 2] = ( β + β ) / 2 = β . (b) Linear but not unbiased. (c) The estimator is linear because it is a linear combination of the response observations, and unbiased because E (3 y 22 - y 23 - y 24 ) = 3 α 2 - α 2 - α 2 = α 2 . (d) The median is not linear in the observations because there is no a such that the estimator may be written as a 0 y , but it is unbiased if the distribution is symmetric. (e) The estimator is linear in the observed responses with a = ( - 1 / ( x n - x 1 ) , 0 ,..., 0 , 1 / ( x n - x 1 )) 0 . The expected value of the estimator is E ± y n - y 1 x n - x 1 ² = " E ( y n ) - E ( y 1 ) x n - x 1 # = " ( β 0 + β 1 x n ) - ( β 0 + β 1 x 1 )) x n - x 1 # = β 1 . So the estimator is also unbiased.
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Unformatted text preview: (f) The estimator is not linear because there is no a such that the estimator may be written as a y . The estimator is unbiased, because of the symmetry of the error density. (g) The trimmed mean is not a linear estimator because of the max and min terms. It is unbiased if the distribution of i is symmetric. Answer 2 (a) yes (b) no (c) no Answer 3 (a) If we do the integral to compute the variance associated with the density, we get a = 2. (b) The BLUE is the least-squares estimator, because the conditions of the Gauss-Markov theorem are met....
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