midterm - ORIE 6700 MIDTERM, FALL 2010 Rules: Use only...

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Unformatted text preview: ORIE 6700 MIDTERM, FALL 2010 Rules: Use only class notes and the one and only course text. You must work on your own without collaborating. Giving help is as serious a form of cheating as requesting help. (1) Let X 1 ,...,X n be a random sample from f ( x | ) = 2 xe- x 1 (0 , ) ( x ) where > 0. (a) Name this density. (b) Compute I ( ) = E h- 2 2 log f ( X i | ) i . (2) Suppose that X i iid N ( , 1) for i = 1 ,...,n where 0 1. Suppose that we are estimating with squared error as the loss function. (a) Find the risk functions for T 1 = X , T 2 = X/ 2, and T 3 = 1 / 4 + X/ 2. (b) Calculate the Bayes risk of T 3 under the prior ( ) = 1 { [0 , 1] } (uniform on the range 0 to 1). (3) Suppose that X i iid Discrete( , 2 , 3 , (1- 6 )) for i = 1 ,...,n where 0 < < 1 / 6. (a) Find I ( ) = E h- 2 2 log f ( X i | ) i (b) What is the Cramer-Rao bound for for the variance of unbiased estimators of...
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This note was uploaded on 12/14/2010 for the course ORIE 6700 taught by Professor Woodard during the Fall '10 term at Cornell University (Engineering School).

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midterm - ORIE 6700 MIDTERM, FALL 2010 Rules: Use only...

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