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midterm - ORIE 6700 MIDTERM FALL 2010 Rules Use only class...

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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 | θ ) = θ 2 xe - θx 1 (0 , ) ( x ) where θ > 0. (a) Name this density. (b) Compute I ( θ ) = E - 2 ∂θ 2 log f ( X i | θ ) . (2) Suppose that X i iid N ( μ, 1) for i = 1 , . . . , n where 0 μ 1. Suppose that we are estimating μ 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 π ( μ ) = 1 { μ [0 , 1] } (uniform on the range 0 to 1). (3) Suppose that X i iid Discrete( θ, 2 θ, 3 θ, (1 - 6 θ )) for i = 1 , . . . , n where 0 < θ < 1 / 6. (a) Find I ( θ ) = E - 2 ∂θ 2 log f ( X i | θ ) (b) What is the Cramer-Rao bound for for the variance of unbiased estimators of θ , the probability of observing the first outcome? (4) (Poisson regression) Let Z 1 , . . . , Z n be fixed constants and let X 1 , . . . , X n be independent Poisson
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